While I do like Skyrim's introduction, I much prefer Morrowind's. In Morrowind it was short and sweet, see a dream sequence (That was skipable! *gasp*) name your guy, tell the guard where you're from, make your class with Socius and then bam exit the building(after stealing everything not nailed down of course

) to Seyda Neen and do whatever you want. It was much quicker when making new characters as opposed to the unskipable movie we are forced to watch each time we create a character in Skyrim. I also really liked Oblivion's start simply because Jean-Luc Picard's awesome voice was used for the Emperor

From an "artistic merit" standpoint, Skyrim wins Hands-down
Skyrim's intro, http://www.youtube.com/watch?v=Q4Y2oO9oZwo, really sets the mood for the game. The tutorial's also pretty good at introducing new players to the mechanics and is not as intrusive as Oblivion's. And for engaging the player AND character, it does everything
Oblivion's does, but better, to get you into the action and judge your character. The biggest dissapointment to me is that neither Hadvar nor Ralof become marriable companions.

I love them both so much.
Morrowind's music, while nice, doesn't fit the intro as tightly, with the cutscene being too "generically" atmospheric, and the Main theme, starting as soon as you "awake", doesn't quite combine with the on-screen events to create such an emotionally-compelling moment.
Oblivion's introduction pales in comparison's to Skyrim because of the lack of unique music - Generic dungeon theme, with generic combat music throughout. It also falls down once you get out of the caves, becoming a case of "Watch how awesome these people are."
From a strictly gameplay perspective, I like Oblivion best.
Oblivion wins because of the "Save at Sewer Exit" and double-confirmation of your choice, allowing you to check out all the races/appearances and benefits of the skills, and creating a new character's a breeze once you're familiar with the game: Just load the sewer exit. It also comes with the best "Starting Packages", giving access to whichever class of weapons and armor you favor, as well as the basic "Starting Spells" in any magic schools you choose. The only problem is with the inequality of gear: The blades you get access to are superior to the Blunt weapons you can use - having access to a Steel Waraxe and Iron Greatsword could have evened that out a bit more.
Skyrim and
Morrowind are somewhat tied: Morrowind gets you into the game "Faster" because it's shorter in terms of duration... but it tends to "feel" longer because it's slower: Once you gain control of your character in Skyrim, you can get through the intro at your own pace. You have to slog through too much conversation to go through
Morrowind's... and the time it takes to loot the Census and Excise Office slows it down more. But, you don't
really need to do that. For "starting packages", while you start in town and can thus buy what you want, the inventory can frequently be underwhelming.
Skyrim, on the other hand... I'm so glad the Stormcloak soldier decides to just get on with getting his head lopped off... After that conversation (Which seems to drag on forever on your not-first character) and sequence, though, it immediately gets better by allowing you to self-pace until you enter the keep and have to wait for the guy to shut up, decide you're standing in just the right place, and unbind you. At which point you take the lead again. And for non-mages, the starting package is superior to
Morrowind's - it lets you play around with the different styles before committing to your favorite loadout that matches your style. While I don't like the gear discrepency (You can't get any decent heavy armor if you side with Hadvar, and I don't like siding with Ralof), and the lack of spells to play around with in the start is underwhelming (Did we really need a
Sparks tome when we already have
Flames to play with Destruction? Why couldn't it give us "Conjur Familiar" to play with Conjuration, and/or a novice Alteration spell such as Oakflesh to test that school out?), I find the free access to different playstyles to make up for that.
It also avoids two of the pitfalls of
Oblivion's character creation by never taking character control away once it's established, and ensuring you're a contributing member of the introduction sequence.