This is a list of well known Skyrim Bugs/Glitches and their solutions. It's not a complete one, but I'll try to update the list as I remember more and also on other people's input. The bugs/glitches listed below should give you an idea what the list is all about.
Disclaimer: The list mostly covers the bugs that are related to game mechanics and the engine and does not include quest, dialogue and other bugs related to the game content in Skyrim.
Inverted SHIFT key function
Description: When you hold down the SHIFT key, your character is walking instead of running, because the function of the SHIFT key has been inverted in the game.
Solution: To revert the SHIFT key function back to normal, open your inventory by pressing I and then close it again by pressing SHIFT+I. Opening another menu and closing it while holding down the SHIFT key does the job too.
How to Avoid: Try not to activate anything (open load doors, start dialogue with NPCs) while holding down the SHIFT key.
Mod Solution: See here.
You cannot quit Skyrim using ALT+F4 keys
Description:You may have noticed that you can not quit Skyrim using the ALT+F4 key combination.
Solution: ?
How to Avoid: NA
Mod Solution: See here.
Windows mouse pointer appears in the game after ALT+TAB (aka. double cursor)
Description: You see the Windows mouse pointer along with Skyrim's mouse pointer when you ALT+TAB in and out of Skyrim.
Solution: ?
How to Avoid: Open the console by pressing ~ key before doing ALT+TAB.
Mod Solution: See here and here.
Low prices for player enchanted items
Description: Despite being a master enchanter you get very low prices for the items you have enchanted.
Solution: The low prices you see do not reflect the actual values of these items. Save your game, quit Skyrim and reload the save in order to get the actual values/prices for them. Yes, this is a bug, not a game feature.
How to Avoid: ?
Mod Solution: ?
CTD upon casting "Bound Sword"
Description: Skyrim crashes upon casting "Bound Sword" with your right hand.
Solution: To fix this use player.resetinventory console command (read more here).
How to Avoid: ?
Mod Solution: ?
CTD while saving the game
Description: The game crashes to desktop when you attempt to save the game. Often happens with script heavy mods and/or mods that save additional data along with the normal save data, with support of a SKSE plugin (a DLL file).
Solution: ?
How to Avoid: Waiting a few seconds before clicking on [NEW SAVE] helps.
Mod Solution: ?
CTD on showracemenu
Description: The game crashes to desktop when you use showracemenu console command.
Solution: ?
How to Avoid: ?
Mod Solution: See here.
CTD while loading a save after game launch
Description: The game crashes to desktop when you attempt to load your save for the first time after you've launched the game.
Solution: ? (Not a solution, but as a workaround, try starting a new game, or load an earlier save that works and then load the save you want).
How to Avoid: ?
Mod Solution: ?
Player's eyes glowing blue (aka. draugr eyes) or other strange FX attached to player
Description: At a certain point in the game you may become infected with a strange FX such as the draugr eyes effect which makes the PC eyes glow blue, or a steam FX from a Dwarven Centurion etc. (Some links discussing the issue: 1, 2, 3.)
Solution: Try using player.dispelallspells console command or using player.addspell <SPELL_ID> add the related spell (see below) to the PC and then remove it using player.removespell <SPELL_ID>.
SPELL_ID for blue Draugr Eyes FX: F71D1
SPELL_ID for Dwarven Centurion steam FX: 30FBE
SPELL_ID for ice wraith FX: CAF8A
See here for more detail.
How to Avoid: ?
Mod Solution: ?
Player's eyes become glitched/bugged
Description: Sometimes your character's eyes may become glitched after using invisibility spells, or after sleeping etc.
Solution: ?
How to Avoid: ?
Mod Solution: There are several causes of this problem and several solutions you might want to try.
To fix the eye glitch caused by invisibility spells:
Eyes Glitch Fix by bellwood2206
Invisibility Glitch - Eyes Fix by KingsGambit
To fix the eye glitch caused by sleeping:
Try using the eye meshes and textures that come with Fair Skin Complexion by HHaleyy.
Player character no longer blinks
Description: You may have noticed that after the last official Skyrim patch (1.9.32.0) your character's eyes no longer blink. This is one of the things that the last official patch broke.
Solution: ?
How to Avoid: ?
Mod Solution: Use either MFG Console or the Sexlab Animation Framework which comes with MFG Console. You don't need to use both.