It is known that setting the secret key to a fixed, public value does not make MACs like CBC-MAC or GMAC into secure unkeyed cryptographic hash functions that could be used - for instance - for digital signatures.
In other words, the resulting hash function is not 1st pre-image, 2nd pre-image, and collision resistant.
What are the MACs that do not suffer from that limitation? I believe HMAC is one of them, provided the underlying unkeyed hash is strong.
Are there any others?
EDIT: if the hash used for HMAC is broken, also the HMAC constructed with it will probably be so.