  The set of dynamic symmetries of the scalar free Schr\"odinger equation in d
space dimensions gives a realization of the Schr\"odinger algebra that may be
extended into a representation of the conformal algebra in d+2 dimensions,
which yields the set of dynamic symmetries of the same equation where the mass
is not viewed as a constant, but as an additional coordinate. An analogous
construction also holds for the spin-1/2 L\'evy-Leblond equation. A N=2
supersymmetric extension of these equations leads, respectively, to a
`super-Schr\"odinger' model and to the (3|2)-supersymmetric model. Their
dynamic supersymmetries form the Lie superalgebras osp(2|2) *_s sh(2|2) and
osp(2|4), respectively. The Schr\"odinger algebra and its supersymmetric
counterparts are found to be the largest finite-dimensional Lie subalgebras of
a family of infinite-dimensional Lie superalgebras that are systematically
constructed in a Poisson algebra setting, including the
Schr\"odinger-Neveu-Schwarz algebra sns^(N) with N supercharges.
  Covariant two-point functions of quasiprimary superfields are calculated for
several subalgebras of osp(2|4). If one includes both N=2 supercharges and
time-inversions, then the sum of the scaling dimensions is restricted to a
finite set of possible values.
