I was watching Prisoner of Azkeban just now and a thought struck me:
Wouldn’t Sirius need a wand in prison to change into his dog form as Peter Pettigrew needed a wand under the Whomping Willow to morph back inot a rat? Did Sirius have one, get one or did I forget it?
You don’t need a wand to transform. Wormtail went for the wand, Harry disarmed him, and then Wormtail transformed.
I’ve found a (minor) plothole, confirmed in book 6:
In book 1, Hagrid mentions that there never was a wizard who went bad that wasn’t a Slytherin. In book 3, we’re introduced to Sirius Black, considered to be one of the worst of the worst (till he was proven innocent, but Hagrid couldn’t know that at the time.) In book 6, it’s confirmed that Sirius was a Gryffindor. Given the personal contact Hagrid had had with Sirius - he was one of the very last people Sirius talked to before the curse thrown by Wormtail - and the anger he had toward Sirius’s betrayal, it’s something of an oversight for him to say all evil wizards came out of Slytherin. He took it too personally.
EleventyOne, I was never under the impression that that statement was meant to be taken literally–just a bit of useful hyperbole when dealing with an impressionable kid. Hagrid is not one of those people (like, say, Hermione) who can be depended upon to be strictly accurate in factual matters.
What always got me about Book Three was this:
The Marauders Map. Okay, so Filch confiscated it and Fred and George found it. But how in the world would they have known how to use it? “I solemnly swear I am up to no good” isn’t exactly intuitive, and it’s not likely that there was a Post-It note stuck to it with the necessary instructions. So what gives?
I think that falls under “plot required, no explanation necessary” category. 
Going back to the OP, from what I understand, an animagus can transform without a wand. And, for that matter, in Sorcerer’s Stone, McGonagall does her transformation from cat to teacher without any wand necessary. Personally, I’ve always gotten the impression that some magic takes wands (charms, etc.) and some doesn’t (apparition). Wands seem to be more of a focus than a requirement for certain magic.
Rowling somewhat gives an answer in her latest interview upon the release of Book 6:
http://www.the-leaky-cauldron.org/extras/aa-jointerview3.html
[spoiler]MA: How did they figure out how to work the map?
JKR: Don’t you — well. This is how I explained it to myself at the time, and this does sound glib. Don’t you think it would be quite a Fred and Georgeish thing to say in jest, and then see this thing transform?
MA: Yeah.
JKR: Can’t you just see them?
ES: But the exact word combination? Is that just a lot of luck, or Felix Felicis —
JKR: Or, the map helped.
MA: Yep, yeah. You can see them sort of answering and joking with each other —
JKR: And the map flickering into life here and there when they got closer and closer, and finally they hit upon the exact right word combination and it just erupts. [/spoiler]
Just like some spells can be executed without a spoken word, so also can they be executed without a wand. I think a wand helps concentrate your power, but it isn’t completely necessary for all magical spells at all times, if you’re a sufficiently skilled witch or wizard.
That’s interesting. I always sort of assumed it was passed down from the original Mauraders to “junior” Maurauders, and so on, over time, in much the same way that Harry himself receives it in the first place. All involved were in Gryffindor after all, and the map is only useful around Hogwarts and neraby environs (Hogsmeade).
Actually, if you will recall, F & G stole the map from Filch’s office. Therefore, they would have had no access to the instructions. However, (and in spite of what JKR herself said!) I would think that intention alone might be enough to awaken the map. F & G have plenty of that!
Before Hagrid’s arrival on Harry’s birthday in Book 1, Harry did things to the window glass at the zoo to allow the huge snake to escape, and to cause Dudley to fall in and end up imprisoned behind the glass. No wand.
Seems that sometimes being wandless renders a wizard powerless, but at other times a wandless wizard can still do stuff.
I’m pretty sure that JKR has said somewhere that wands are required for controlled magic. I’m pretty sure that most(if not all) young wizards and witches have incidents similar to Harry’s.