  Muttalib-Borodin determinants are generalizations of Hankel determinants and
depend on a parameter $\theta>0$. In this paper, we obtain large $n$
asymptotics for $n \times n$ 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 characteristic polynomial, several
central limit theorems, and some global bulk rigidity upper bounds. Our results
are valid for all $\theta > 0$.
