The salt may be considered public practically by definition.
As for what hash to use and how "safe" it is, I would stick with the recommendations NIST makes for U.S. government agencies. As the Wikipedia article says:
SHA-1 is being retired for most government uses; the U.S. National Institute of Standards and Technology says, "Federal agencies should stop using SHA-1 for...applications that require collision resistance as soon as practical, and must use the SHA-2 family of hash functions for these applications after 2010"
You can speculate about how safe or unsafe various algorithms are, but why bother when you have a spec the U.S. government considers good enough for its own use?
By the way, your question appears to reflect a misunderstanding about how these analyses are performed. There is much more to a hash function than its number of bits.
Brute-forcing a 160-bit hash like SHA-1 requires 2^160 effort; that is simply not going to happen, no matter how many "GPUs" you throw at it. In fact it probably will not happen in the lifetime of the universe. (True, brute-forcing a collision requires "only" 2^80 effort. But it also requires 2^80 storage. Good luck with that.)
The reason MD5 and SHA-1 are considered "possibly unsafe" is that weaknesses have been discovered in each algorithm that might reduce the work way, way below the brute-force effort. The reason for switching to a newer algorithm (like SHA-2) is to avoid those known weaknesses; the reason for using 256 bits is to provide some buffer against unknown weaknesses. It has nothing to do with "GPUs"; to resist brute-force attacks, 160 bits is plenty.
Similar statements hold for all cryptography, by the way.
Now, it is impossible to overstate how much more the folks at NSA know about this stuff than you or I do. They make recommendations for U.S. government agencies and U.S. industry via NIST. So unless your adversary is itself a major world government, the NIST recommendations are more than sufficient. Ignore any self-proclaimed "experts" who try to tell you otherwise; NSA is better at what it does than you can possibly imagine.
If your adversary is a major world government, nobody at SO is qualified to help you. And you have much bigger problems than which hash function to use.
Of course I am assuming you are already using this function as part of a standard password hashing scheme, like PKCS#5 (also known as "PBKDF2"), and your question was purely about what hash function to use.
PBKDF2 recommends a minimum of 1000 iterations, but you can choose whatever you want. The attacker's brute-force effort goes up linearly with that number, so 10000 iterations vs. 1000 means it will take them 10 times longer to brute-force each password.