  The paper is on the vanishing topology of singular Milnor fibres of
holomorphic families of arbitrary square, symmetric and skew-symmetric matrices
with sufficiently many parameters. We define vanishing cycles on such fibres,
prove an extended form of the Damon-Pike $\mu=\tau$ conjecture about the
families of a special type, and make first steps towards understading of the
monodromy of matrix singularities. We also prove a Lyashko-Looijenga type
theorem for simple matrix families, and point out a surprising relationship
between certain Shephard-Todd groups, simple odd functions and a sporadic part
of the Bruce-Tari simple matrix classification.
