  We study a class of one-matrix models with an action containing nonpolynomial
terms. By tuning the coupling constants in the action to criticality we obtain
that the eigenvalue density vanishes as an arbitrary real power at the origin,
thus defining a new class of multicritical matrix models. The corresponding
microscopic scaling law is given and possible applications to the chiral phase
transition in QCD are discussed. For generic coupling constants off-criticality
we prove that all microscopic correlation functions at the origin of the
spectrum remain in the known Bessel universality class. An arbitrary number of
Dirac mass terms can be included and the corresponding massive universality is
maintained as well. We also investigate the critical behavior at the edge of
the spectrum: there, in contrast to the behavior at the origin, we find the
same critical exponents as derived from matrix models with a polynomial action.
