Muttalib-Borodin determinants are generalizations of Hankel determinants and depend on a parameter theta > 0. In this paper, we obtain large n asymptotics for nxn Muttalib-Borodin determinants whose weight possesses an arbitrary number of Fisher-Hartwig singularities. As a corollary, we obtain asymptotics for the expectation and variance of the real and imaginary parts of the logarithm of the underlying characteristic polynomial, several central limit theorems, and some global bulk rigidity upper bounds. Our results are valid for all theta > 0.