Call Bear / Wolf Cheat
When you have received the Call Bear/Wolf ability, summon one of the animals and then rest. When you wake up the animal will be gone, but if you load the autosave just before you rested, the animals you summoned will be permanent. This glitch, however, will not affect most other summoned creatures, with the exception being the Centurion Sphere created by the "Dwemer Animunculi" spell.
If you kill Ingmar when he is at the Valbrandr Barrow, his body miraculously teleports to his house in Skaal Village, with all the equipment on him. Note that this was tested on Xbox:GOTY and he was INSIDE the barrow when he was killed.
Werewolf's head is gone
While playing as a werewolf the werewolf's head will disappear while underwater. (This was fixed in a later patch)
Can't finish the Discovery in the Mine quest
If you have very high personality or a charm on when you go to the island to speak with Graring and if you allied with Falco, Coventina Celata won't attack you or Graring. Otherwise Graring will just say "put down your fists (race) unless you want to fight". You have to find and kill Conventia Celata, who should be behind a rock next to the river. Your journal will be updated allowing you to finish the quest.
Can't finish Betrayal at Brodir Grove quest
If you do not speak with Bereditte Jastal before becoming the chieftain of Thirsk to get his dialogue: "My name is Bereditte Jastal. Perhaps you'd like to buy one of my books? I'm a bookseller by trade, but my real passion is writing. I recently finished two scholarly works, Thirsk, a History and Sovngarde, a Reexamination. I have copies of both for sale, among others.", you will not be able purchase the book "History of Sovngarde" and thus you cannot complete the quest.
Werewolves and Permanent Fortify Attributes
Every time before you change into a werewolf, the game saves your stats, no matter what they are at the time. If you're under the influence of any Fortify Attribute effects (or Drain Attribute effects for that matter) this is what Morrowind remembers. Once you change into a werewolf, regardless of any spell or potion effects, your attributes are set to the proper levels for werewolf form. Werewolf form lasts for nine hours if it's the infectious kind or six hours if you're using Hircine's Ring. During this time, the Fortify Attribute effects may wear off, depending on the duration of the potion or spell. Unfortunately, this could mean having one of your stats plummet to zero while in werewolf form. If you then wait out the transformation, the game had saved your stats before you changed so it "resets" them back to these levels, even if they're fortified. This means you'll have fortified attributes but no longer be under the influence of any Fortify Attribute effect. In other words, the stat changes are permanent.
There are very obvious ways to abuse this in order to gain permanent attribute or skill increases: by drinking potions or casting spells right before you change, such that they wear off while you're a werewolf. However, it can break the game if you've raised your attributes to excessive levels. It also makes it impossible to see your base stats until you level up. At this point the only way to reset your stats to normal is to use the SetAttribute command from the console.