Is there a key for AES-128, 256 etc, where cleartext equals ciphertext?

In the "ideal cipher" model, the block cipher is a permutation of the space of input blocks, chosen uniformly among all such permutations. A plaintext that gets encrypted to itself is a fixed point for the permutation; it is expected that about 63.21% of all permutations have at least one fixed point (a permutation with no fixed point is called a derangement).

Thus, assuming that AES is an ideal cipher, then it is expected that for about two thirds of possible keys, there will be at least one plaintext block that is encrypted to itself.

Now, finding that fixed point is quite another thing; we in fact expect not to be able to easily find that fixed point or even ascertain whether it exists or not -- the "moral" reason is that finding a fixed point on a random permutation requires hitting it exactly, which is akin to a brute force on the whole input space, of size 2¹²⁸ in the case of AES. In other words, if we could find fixed points or even prove (non-constructively) their existence or non-existence for any specific key, then this would mean that we could differentiate AES from the ideal cipher model, and that would be worrying.

(See also that answer on crypto.SE.)

Note: while a majority of AES keys should have at least one fixed point, it is strongly expected that there is no key that turns AES into an identity function (i.e. all plaintexts being fixed points): there are 128! possible permutations, a number waaaay larger than the 2²⁵⁶+2¹⁹²+2¹²⁸ possible AES keys, thus the identity permutation is very unlikely to correspond to a key. But there again, we don't expect to be able to prove it.