Neat question. We think that the answer is that some superconductors
are magnetic. However, most aren't. The reason is that
superconductivity arises from the behavior of pairs of electrons, where
electrons of opposite spin (and hence opposite intrinsic magnetism)
pair up. If the state they pair up into has no orbital magnetism (i.e.
no circulating current) then there can't be any magnetism without
breaking the superconducting pairs and losing superconductivity. (In
Type II superconductors, magnetism can penetrate the superconductor,
but only in little vortices within which the superconductivity is
lost.)
However, it is now strongly suspected that there can be types of
electron pairing in which the pairing state does have circulating
currents, and is intrinsically magnetic. Something very like that is
already known for superfluidity in helium. Superconductivity of that
type would require magnetism. In any particular sample, it would just
be an accident which way the magnetism pointed.
Mike W.
For the first kind of superconductor Mike mentions, you can make a
superconducting material stop superconducting by introducing magnetic
atoms in it. Superconductors normally expel magnetic fields from their
bulk, called the "Meissner effect". Introducing magnetic impurities to
a material which otherwise superconduct makes the effect go away. Here
is a cool link from Lawrence Berkeley Labs about the different kinds of
superconductors:
Fun article at LBL Tom
(published on 10/22/2007)