Objectives in the Underworld
The Underworld in Ultima V is very big, and also very empty. It is a very dangerous place, and there are only certain tasks to be done, so here a solution with maps of exactly what to do. Note that before even entering the Underworld, the magic carpet, the grapple and the spell blink are absolutely essential for making any progress.
Underworld under Despise
No map is included, since this is the easiest section with no obstacles.
After leaving dungeon Despise, just jump onto the carpet and fly over the water, until you've found the Ararat; don't bother with searching inland. Once there, talk to Captain Johne, and if you want, let him join the party. After that, leave the way you came from. This area can also be entered by sailing a ship into a whirlpool.
Underworld under HythlothShard of Cowardice found here, but also the Mystic Weapons and Mystic Armour. This however, is also the most dangerous part of the Underworld.
After exiting Hythloth, first go to the volcano of the Abyss. In the middle, you can find all the Mystics by searching, although it damages you. Afterward, follow the line on the map. The grapple is useless, so the spell Blink has to be use at the yellow parts of the line, while the carpet is needed to get over the swamp.
Underworld under DeceitShard of Falsehood.
While there is a long way, which involves falling down many waterfalls, there is also an easy way. Follow the line on the map and grapple over the marked mountains on the map, then get the the lake in the west. Jump on your carpet and then get the shard from the island, without any problems.
Now backtrack back to Deceit and exit.
Underworld under Wrong and CovetousShard of Hatred.
This section of the Underworld can be entered from either dungeon Wrong or dungeon Covetous, note however, that the way from Covetous to the shard is longer and more dangerous, since it also involve using the grapple. Once you've reached the start of the purple line, use the grapple to (k)limb this long distance (you will take damage), until finally reaching the shard at the center.
Leaving is possible with Blink. Then exit through the dungeon of your choice, although Wrong is easier to reach.
Underworld under Destardknights and Lord British's Amulet.
There are two ways to get here. The straightforward one is to descend dungeon Destard, while the unusual one is, to use a skiff or magic carpet to enter the waterfall in Spiritwood. One way or another, once there, follow the line and use the carpet to glide over the water and to avoid the swamps. After the big swamp, you should easily find the amulet in the east.
Afterwards, return to the forechamber and blink north, then return to Destard to exit.
Underworld under ShameDoom.
After exiting Shame, fly east with the carpet, then blink at the marked position to enter the central underworld. Fly over the rough seas (damage) and then over the lava in the center island (even more damage!) to enter the darkness. Equip the amulet to see the entrance to Doom. Yell "Veramocor" to open it, and rest up beside it (this area is safe) until you are ready, then descend into it.