  We consider Lie superalgebras under constraints of Hamiltonian reduction,
yielding finite $W$-superalgebras which provide candidates for quadratic
spacetime superalgebras. These have an undeformed bosonic symmetry algebra
(even generators) graded by a fermionic sector (supersymmetry generators) with
anticommutator brackets which are quadratic in the even generators. We analyze
the reduction of several Lie superalgebras of type $gl(M|N)$ or $osp(M|2N)$ at
the classical (Poisson bracket) level, and also establish their quantum (Lie
bracket) equivalents. Purely bosonic extensions are also considered. As a
special case we recover a recently identified quadratic superconformal algebra,
certain of whose unitary irreducible massless representations (in four
dimensions) are "zero-step" multiplets, with no attendant superpartners. Other
cases studied include a six dimensional quadratic superconformal algebra with
vectorial odd generators, and a variant quadratic superalgebra with undeformed
$osp(1|2N)$ singleton supersymmetry, and a triplet of spinorial supercharges.
