  We analyze whether Ulrich modules, not necessarily maximal CM
(Cohen-Macaulay), can be used as test modules, which detect finite homological
dimensions of modules. We prove that Ulrich modules over CM local rings have
maximal complexity and curvature. Various new characterizations of local rings
are provided in terms of Ulrich modules. We show that every Ulrich module of
dimension $s$ over a local ring is $(s+1)$-Tor-rigid-test, but not
$s$-Tor-rigid in general (where $s\geq 1$). Over a deformation of a CM local
ring of minimal multiplicity, we also study Tor rigidity.
