Is there any bias whatsoever in modern hash function outputs?
If there were any detectable bias, I wouldn't be posting about it on a forum of pseudonymous wackos like me on the internet—I would be getting top billing (or beaking, as the case may be) in a top-tier cryptography conference, and the champagne would be popping, and Twitter would be abuzz with speculation, and Hacker News would be extra insufferable.

Specifically, with the exception of 2-pass Snefru[1][2], no major hash function with advertised preimage and collision resistance has ever seen its preimage resistance broken[3] (archive).  (Yes, there's a paper that everyone and their dog cites on an MD5 preimage attack[4], but it's not cheaper than the best generic attack[5].)



P.S.  The formula $\epsilon = 2^{-(s n - k)/2}$ appears to be a quotation from an ID Quantique marketing whitepaper with a garbled definition which Paul read out of context to draw a nonsense conclusion in the self-accepted answer of his own that he cited.