Do well-known hash functions have any "impossible" output values? What I'm wondering is whether the codomain and image of common hash functions are strictly equal… or whether there are any known values which have no possible preimage.

For example, SHA-1 outputs a 160-bit digest. Can it be proved that every possible sequence of 160 bits is a possible output of SHA-1 (even though corresponding preimages cannot be easily computed, by design), or are there certain outputs which can be demonstrated to be impossible based on the construction of the hash function?

This question about uniformity of hash function output seems related, though what I'm interested in has less to do with overall uniformity, and more with provable existence/non-existence of any preimage for specific outputs.
No, it is not known that any cryptographic hash in common use has impossible output values (that is a bitstring of the appropriate size to be an output, but that is not reached by any input message). That's at least for MD5, the SHA(-1/-2) family of FIPS&nbsp;180-4, SHA-3 of FIPS&nbsp;202, RIPEMD-xxx, Blake/2, Tiger/2, Whirlpool.

In fact, it would be surprising that such impossible output value exists, and even more that it could be exhibited:


For a hash modeled as a random function, the probability that there is such value is lower than $1/2$ for a $n$-bit hash when there are more than $n\,2^n$ inputs; and that probability lowers with larger message input sets. See this for details.
For a Merkle-Damgård hash with Davies-Meier round function, the message block at most twice as large as the output, and the length padding at most about a quarter of the block size (like MD5 and direct larger successors are), under the assumption that the underlying block cipher is a Pseudo Random Permutation, and by an adaptation of the above reasoning, it is very probable that all outputs are reached with a single message block; and even more probable that they all are reached with two message blocks. Anything else seems to require a severe defect of the underlying block cipher, and any known such defect seems considerably lesser.
For the sponge construction of SHA-3, a similar argument can be made that a severe defect of the sponge function would be required to have some output values unreachable.