Tough travelling feature has moved to the Fantasy Hive.
This month the theme is shapeshifters: “Shapeshifting is frequent among both WERES and MAGIC USERS. The usual form taken is that of a wolf, but lions, eagles, serpents, owls and cats are common too. In all cases the Rule is that the Shapeshifter cannot stay too long in ANIMAL form without actually becoming that animal and losing touch with her/his human thoughts.”
– The Tough Guide to Fantasyland by Diana Wynne Jones

Most shapeshifters that I’ve come across can shift into only one form, for example:
Wolves
Werewolves are probably the most common shapeshifters. For example, Kitty from Carrie Vaughn’s urban fantasy series.

Bears
For example, Beorn by J. R. R. Tolkien. He’s suspicious of the twelve dwarves but ends up helping them.

Dragons
I think that all shapeshifting dragons I’ve come across are in fact dragons who can change into a humanoid form, not the other way around. The Dragonlance universe has several of them and so has the Invisible Library series by Genevieve Cogman.

Elves
Most elves can appear to be human so that humans don’t notice them. For example, Toby Daye by Seanan McGuire.

But there a few creatures which can change their shape into more forms:
Various gods and tricksters
Most, if not all, mythologies have shapeshifters. Perhaps the best known is Loki from the Norse myths who goes so far that he even bears a foal when he’s a female horse. Odin is another one who likes to change his shape frequently. So is Zeus and many of the other Greek (and Roman) gods.

Dracula
He can change into a wolf, a bat, and even mist.

The D&D Druids
They can take the forms of most animals a few times a day (depending on the level). However, I must confess that the only druid character in fiction I can think of is Jaheira from the Baldur’s Gate games. Surely there must be more than that? (yes, I’m currently playing a high-level druid in a table-top campaign :))

Kandra
Kandra are creatures who can take on the appearance of another, human or animal, by digesting their flesh and bones. They appear in the Mistborn series by Brandon Sanderson.

Changelings
In Star Trek: DS9 we have an entire species of beings who can take pretty much any form they want. Odo, of course, is the most prominent such character in the show.

Marvel comics are also pretty full of shapeshifters.
Most of them are superpowered people who change into their powered form and back to their normal selves. For example, the Hulk, She-Hulk, Colossus, or Wolfsbane (Marvel’s werewolf).
However, they also have the Skrulls, a species which can shapeshift to any person.
Perhaps the most known Marvel shapechanger currently is Mystique who can take the form of any person.