The fact is I'm not quite sure if my question statement is true, however all the MAC constructions I know of (e.g. CBC-MAC, CMAC, HMAC) are deterministic, whereas many constructions for digital signature are randomized (e.g. DSA, ECDSA, RSASSA-PSS). I also know that RSASSA-PKCS1.5 is deterministic but still considered secure (though not recommended in favor of RSASSA-PSS).
Why is it the case that most MACs are deterministic whereas most digital signature schemes are randomized? Are there randomized MACs that I simply don't know about, and do they provide any advantages over the deterministic approach? Is there some fundamental difference between MACs and digital signatures that makes the randomization more desirable (but still not required) for signatures?
Are there randomized constructions for MACs or digital signatures that breaks badly if the randomizer output is not indistinguishable from random, or do they all simply reduce to a still secure deterministic scheme?