Asgard and Vanaheim-
Preparing the body for burial in Asgard is a very solemn duty performed by the closest member of the dead's family. If a spouse exists the body is prepared by them. To prepare the body, it is cleaned and treated with special oils and perfumes before being redressed into the dead's finest clothing or armor. If the Asgardian has a favored weapon it could also be buried with them, however if it is particularly powerful (such as Mjolnir) it would be passed onto someone else. Craftsmen could also be buried with their tools.
Asgardians have two different methods of burial for their dead. The, by far, older of the two methods was the use of burial mounds known as barrows. The bodies of the dead would be cremated and put in urns before being buried with the most important worldly possessions of the deceased. Most middle/lower class families would use one barrow for many generations as the maintenance and initial construction costs of a barrow could be expensive. The poorest of Asgardians would not even use barrows at all and would take the cremated remains of loved ones and bury the urns around other barrows (usually of dead heroes or other such important Asgardians) and indicate the spot with a small mound of stones. The Borson Barrow is the largest in Asgard and houses the remains and possessions of Vili and Ve, Odin's brothers that died in the fight against Sutur. The area around the barrow is riddled with remains of those that could not afford their own and is the closest thing to a true 'graveyard' that could be identified in Asgard although it is not labeled as such.
Within a barrow everything from food and alcohol to entire boats could be buried with the dead. Weapons, treasures, and even favored horses also tended to be buried although the horses were usually added later as most could not afford to kill a good horse until it became too old or injured in some way. Theft from a barrow is not unheard of but if anyone is caught having done so the punishment is death and dishonor.
The second method of burial in Asgard, being burned on a boat floating towards the edge of Asgard, was brought over by the Vanir. The body is prepared in all of its finest clothing and surrounded by possessions before being set afloat. Under the body is a thick pallet of fuel that helps the fire burn hot enough to properly cremate the body in a short amount of time -something that is not needed in barrow burials as there is no danger of the pyre floating off the edge of Asgard.
The Vanir prefer sea burials opposed to barrow burials due to the importance of water to their culture. Vanaheim is, unlike Asgard, a series of islands and fjords and the realm experiences quite a lot of rain. The largest island of Vanaheim houses the capital and palace and even that island has multiple large rivers running through it. Travel through the realm always requires traversing water in some way, so it seemed only natural that to travel properly to the afterlife it would also require passing over water.
Due to how long it has been since Vanaheim and Asgard have been in alliance with each other the both burial methods are well entrenched in both realms and several individual customs have crossed the boundaries to be included in both methods such as cremation which was not originally a Vanir custom and burying the dead in their finery which was not Asgardian in origin as there was not a physical body being buried -although now the bodies are cremated in their best clothes no matter if a barrow or sea burial is taking place.
Seven days after the burial they would then have a feast to celebrate and honor the dead. During the feast the last will of the deceased is read and the heir formally receives whatever they had been left. The feast itself is a rather small affair compared to others in Asgard and lasts only until the next morning with only those closest to the family involved. The exception to this is for mass funerals after battles or wars, or the death of someone within the royal family. Those feasts last for a week and involve the entire city.
Asgard's Dishonored Dead-
In Asgard, if one commits a grievous enough crime you are not afforded any rituals for death. To be a dishonored dead is not a fate that is carried out too often as even most criminals are allowed to receive at least some modicum of respect after they died. If someone falls into the exception to this, however, there is no burial at all. The worst offenders against Asgard or her people are given the most severe punishments after death and against their lifeless bodies. The body will be torn apart by wolves and dragged through the main streets of Asgard. The wolves will be allowed to fight over and even eat parts of the corpse until they lose interest in it. Once the wolves are done the remaining parts are found and then hung from the observatory so that they fall into Ginnungagap as they rot away.
In Jotunheim, cremation is also a key part of their rituals. However, unlike in Asgard, Jotunheim do not collect the ashes of their dead into urns to bury. Jotnar spread the ashes of their dead in their fields for both spiritual and practical reasons. The idea of their bodies being used to support the living goes all the way back to the legend of Ymir's body being used to form the world -albeit the Jotnar version does not attribute that being done by Odin and that the world made was specifically that of Jotunheim. For practical purposes, the ashes of the dead help to grow their crops that are more nutritious and heartier than those that grow without the added fertilizer which is key when you have such a short growing season.
Jotnar spread the ashes of the dead and pass on their possessions on the same day -nine after the cremation is completed. Prior to the cremation the rituals involve laying the body out in the middle of the town in an ice building constructed purely for that purpose and taken down after. The body remains there for three days in full view though protected behind a layer of clear ice. This three day wait period before cremation is to ensure that the dead is truly dead as the cold temperatures of Jotunheim can often lead to a hibernation state that can fool even other Jotnar. Although this wait period can be abbreviated to a single day if the death is particularly violent/obvious such as a death caused be beheading or impalement or other grievous injury.
The actual act of spreading the ashes is ritual attended by the entire family and a priest to ensure that it is performed correctly. The ritual includes mixing the ashes with special herbs and powdered minerals dug out from volcanic caves before being spread during a hymn that speaks about thankfulness to the spirits and the renewal of life. An incorrectly performed Ash Ceremony is thought to be the cause of plagues within the crops or particularly bad harvests caused by the spirits of the dead being displeased. The immediate family of the deceased gets the first bushel of crops out of that particular field during the next harvest as a show of respect for their loss. Each clan have particular fields that they feel the greatest connection to due to how many of their ancestors have been returned to the soil there and react very aggressively to any attempt to be removed from the proximity of those places. Thereby most of the infighting between clans revolves around the boundaries of ancestral grounds and fields. Even the nomadic tribes have fields that they consider 'theirs' that they use year after year.
Salting a field in Jotunheim is a particularly strong offense as it not only makes extremely valuable crops to die and poisons the ground but because it is also considered an act of aggression towards the ancestors of that particular clan. Even the most deadly and vicious clan blood feuds rarely escalate to the level where one family would damage the land of the other.
Neither the elves of Alfheim or the elves of Svartalfheim cremate their dead. Instead, elves inter their dead whole inside large community barrows. The elves weave silk shrouds to wrap their dead in before burial within the walls of the barrow, upon which they grow pale colored Gladioli in massive groups. Elves of Svartalfheim also sometimes add on a mask to the burial attire for a variety of reasons from a show of respect to an indication of someone without a family to also those that were killed by sickness. The type of mask is how the reason behind its use is determined. Some examples of different styles of masks are: masks of respect are very finely crafted and decorated with precious metals and stones, masks made to indicate sickness are made out of a specific wood and protective runes are carved along the edges, masks for those without family are often all but featureless, and masks done fore religious figures such as priests get decorated with carvings of branches and other plant life.
Once a barrow fills with dead, they simply seal the barrow with a carved stone and then move onto another spot nearby. Barrows can be extended underground however they usually aren't. Two levels is usually the maximum depth of a particularly large Alfr barrow and more typically they only have one. On Alfheim, each community puts their barrows in ever increasing rings around their towns. They see this as being protected by the spirits of their ancestors. Contrarily, Svartalfheim elves place their barrows further away from their homes -sometimes far enough away that to reach them requires a full day's travel.
Elves of both sorts will dress their dead in their finery but will not lay goods out with them, with the exception of perhaps a few pieces of jewelry. At most, incense burners will be place in the wall beside the body so that the mourners that visit can light some for the spirits of the dead. Once a barrow is sealed, it cannot be reopened and there are several spells placed onto the stone door to ensure that grave robbers cannot get in. Any mourners that come to visit their dead loved ones after a barrow is sealed will simply burn incense outside the door or on the earth dome of the barrow.
Of course Elves are not the only inhabitants of Svartalfheim. The dwarves of Svartalfheim create massive mausoleums that house individual stone tombs that line long family wings. The tombs have heavy tops carved into the likeness of the dwarves interred within and sometimes entire statues of the dwarves standing guard in front of the bodies. Dwarves will make intricate mazes of different halls and wings for their dead, resulting in an almost second city beneath the first. If they truly run out of room due to impassable geology or simply the rest of the city being in the way, the Dwarves will build another mausoleum to mirror the location of the first on the other side of the city. Entrances of the mausoleums are often many stories tall and take years upon years of constant carving and craftsmanship to construct.
Before a dwarf is interred in their individual tomb beside their family, they are laid out in large chambers near the entrance to the mausoleum for friends and family to mourn for a full day and night. After the viewing period is done, the body is place along with any tools or weapons they owned into their stone sarcophagus. The lid will be sealed in place with a series of mechanical locks ensuring that nobody can get inside. If a dwarf is older they most likely had the lid of their tomb being carved in their likeness for many years in preparation, however death of the young or those who die in battle usually must have a second carved topper added after their initial burial. In some cases it has taken many decades for a fallen dwarf to receive his likeness carved in the special marble that they use for tombs. Names are not recorded on the tombs and so, if there are multiple sarcophagi without faces it is up to the family to recall whom is buried where so that they will receive the proper likenesses. The few exceptions to this is when a family is intensely poor and cannot afford fully carved statue of their dead or those who have no family to do the remembering. The middle class may choose to only have busts made of their dead but it is still preferable to have a full body likeness constructed.
Dwarven mausoleums are kept open and maintained to near pristine levels by caretakers who will repair and clean and help guide visitors and any other thing that might be required of them. The caretakers often have quarters located at the very entrance of the mausoleums so that they are easily reachable by anyone who might need their assistance. They are the only ones allowed to keep a record of who is interred where as they are in charge of maintaining various family's loved ones at varying levels of attentiveness. The basic maintenance is all done for a fee collected by a tax on the people and anything above that basic level is paid for by the family of the dead out of pocket.
The fire demons of Muspelheim are somewhat unique among the realms in that they have very little in the way of ritual for their dead. Most are buried with very little fuss or concern. Mass graves are commonplace, especially after battles. Fire Giants that still maintain a memory of Jotunheim from before they fled the Jotnar place a bit more significance on death than the majority but even they do not dedicate very much time or effort to venerating their deceased. Most burials in Muspelheim is done solely for the utilitarian purpose of removing bodies from where people dwell. Cremation is popular but little importance is placed on the ashes of the dead unlike in the other realms. Often cremated remains are simply dumped out wherever it is convenient or left within remains of the pyre they were burned on. Possessions are passed on quickly and immediately, although the process can easily be disrupted by those who disregard wills or those that simply never make them. Conflict is often solved very simply by the strongest (or those clever enough to swindle the others) in the argument winning.
Niflheim is a realm cloaked in mystery and the few constant inhabitants tend to be quite monstrous. Trolls and the occasional raiding band of giants seem to be the only intelligent beings that inhabit Niflheim. Any Jotnar that are killed while on Niflheim are taken back to Jotunheim for burial and trolls cremate their dead's bodies very quickly wherever they are and then spread the ashes before moving on. The only permanent societies of intelligent life on Niflheim were refugees from the Giant wars on Jotunheim such as the Rime Giants that nobody has seen or heard from in many centuries. It is assumed that there aren't any permanent residents of the realm any longer. Niflheim tends to have much in common with the concept of limbo wherein nothing is meant to linger there and should quickly pass onto another realm such as Helheim.
Helheim is of course the realm of the dead. As such, Helheim itself does not possess any death rituals.