Which MACs can be converted into a secure unkeyed hash function? 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.
I have ask as part of my answer, "What problem are you trying to solve?"

Do you want a secure unkeyed hash function? If you do, then there are plenty of them around. Even some of the ones that are broken for some uses might be okay for yours (SHA-1 springs to mind -- note the discussion above on HMAC and how broken a hash function has to be). But really, you ought to look at SHA-256, SHA-512, or the new SHA-512/t construction.

If you don't like that, look at any of the SHA-3 finalists. (Full disclosure, I'm the co-author of Skein.)

You're right that HMAC is one of them -- but that's basically the null statement. HMAC is a wrapper around an unkeyed hash function that strengthens it against a number of attacks (and adds a key). If you remove the key (or make it a constant), then all you have is a function with two iterations of a hash function with a bunch of frosting and sprinkles over it. It is obvious that if you start with a secure hash function, make a secure MAC and then remove the key then that you will end up with a secure hash function.

What are you really asking? What do you really want to do?

Jon