<beware potenial for minor spoilers in this thread>
Considering the story and the setting, this game could have come out dead serious. Fortunately that is not even close to the truth. The designers obviously have a wicked, subversive sense of humor. Some of my favorites:
1. For some reason I find the leader of the thieves circle in the Hub to be hilarious. He seems awfully cheery for a guy living in a basement.
2. Dogmeat's singleminded charges lead to some pretty funny occurances. Similarly there is the fun to be had by giving an automatic weapon to an NPC.
3. The obvious glee somebody had creating animation for the blowing apart of bodies is there for all to see.
More?

Almost forgot...
4. Try putting in a character with an intellegence of 1!

Oh man, I remember getting intelligence 1 and talking to Torr in Fallout 2. I almost died from laughter!

Dialogue with Enclave trooper in GECKO power plant. Oh ye, this one is funny too.

The humor in the Fallout series is one of the things that make it so special. I remember laughing quite a bit or being shocked (in a good way) at some of the things I saw or heard. The best part is how it all changes when you change your character stats, adding a great amount of replayability.
It makes it worrysome with the new game in the series about to be released, and if it can even hold a candle to the thought and creativity that went into the originals.

A game by the people, for the people, one could say.
Alternatively, one could say it was a damned funny adventure of awesome proportions.
And my 2 cents; Cassidy saying his Dad named him after a guy from a comic book.
Power Glove, bloody mess, and crits to the eyes.
The Bridge Of Death!

My two cents on the subject, some BIG SPOILERS:
Sure, there's a lot of humor in Fallout, but I for one would like to state that from my personal experience with Fallout 1 & 2, this humor comes mixed in with melancholy and sadness. There are a lot of jokes and funny references, specially in Fallout 2, but parts of it, and most of Fallout 1, take on a rather melancholic tone.
As an example, take the intro to the original Fallout, with its TV full of boastness and happy-go-lucky news of the annexation of Canada and Corvega and Handy-man adverts contrasting harshly with the destroyed landscape, giving off the sense that they just didn't know what was coming.
When the Overseer briefs the player on the urgent need of a water chip, I think there's a worry on the part of the Overseer about the player, if he's going to make it or not, and even a sense of sorry for the player since he got the poor fortune of being the one chosen to brave the Wasteland and save the Vault.
SPOILER FOR FALLOUT
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The ending to Fallout, probably one of the best endings to a game ever, with its unseen choice of shooting up the Overseer or just letting him go, and then watching the Vault Dweller leave the Vault forever always fills me with sadness and rage at the same time. How could they forsake their saviour after all he went through?
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END SPOILER
Another example, the intro to Fallout 2, depicting sick children, dying brahmin, and withering crops, showing off a comunity of simple and honest people that wanted their freedom struggling to survive in the harsh and dead lands of the Wasteland. There's a sense of duty to save this community branded upon the player from the very first minutes.
Also, the sound design. The game's atmospheric sounds were lacking in any sort of 40's or 50's happy music. This was a tough, deserted land and the music and sound depicted it perfectly.
My point in all this is that yes, there was a lot of humor in Fallout, but lets not forget or ignore that there were also feelings of dread (entering the Glow), sadness, anger and irony.

The Humour in Fallout is great :D No doubt about that, i mean it takes itself seriously, but it's just the touches here and there that make it hilarious, like being able to play the game with a completely moronic character.

By no means was I implying that the game is a silly laugh fest. I definitely agree with the post above regarding the melancholy mood. It is very much a serious tale about serious things. I think the unexpected bits of humor and references are there out of necessity to lighten what could have been a real downer of a game. Just like a joke in a serious movie lightens the mood, the humor here balances the downbeat aspects, making a more complete game.

As sorrowful as the story is, as destroyed as the wastes are the humor becomes that much more profound!
That meat on a stick could be anything! In fact... it could be you that the next guy eats!
Even Vault Boy getting his eyes burned out of his skull when the vault door opens for the first time!
So much dark, unforeseen humor.

There are plenty of very funny bits in this game and this makes Fallout series one of the greatest I have ever played. For example, the conversation after staring in the porn movie in New Reno in Fallout 2 :D

Especially if you have the perk for "added random encounters" - I forget what it's called. But with it, I've run into a "roswell" type alien ship - one of the aliens holds a photo of Elvis that is worth 400 caps if you trade it!
Another encounter is with a 1940's era British-style phone booth, that flashes and then disappears - Dr Who!
One of the NPCs says "Lets play Global Thermonuclear War." From the movie Wargames...also I found a different character that asked if I would like to play a nice game of chess!
The Brahmin that don't just moo, but actually go "Moo, I say! Moo!" as a whole sentence crack me up!

Fallout 2 was actually the first game that introduced me to Scientology with the Hubologist parody.

I really found the irony of the water chip in F1 breaking to be hilarious. Of all the things to happen. And notice how nobody else worries about their water supply.