  We provide a systematic study on the possibility of supersymmetry (SUSY) for
one dimensional quantum mechanical systems consisting of a pair of lines $\R$
or intervals [-l, l] each having a point singularity. We consider the most
general singularities and walls (boundaries) at $x = \pm l$ admitted quantum
mechanically, using a U(2) family of parameters to specify one singularity and
similarly a U(1) family of parameters to specify one wall. With these parameter
freedoms, we find that for a certain subfamily the line systems acquire an N =
1 SUSY which can be enhanced to N = 4 if the parameters are further tuned, and
that these SUSY are generically broken except for a special case. The interval
systems, on the other hand, can accommodate N = 2 or N = 4 SUSY, broken or
unbroken, and exhibit a rich variety of (degenerate) spectra. Our SUSY systems
include the familiar SUSY systems with the Dirac $\delta(x)$-potential, and
hence are extensions of the known SUSY quantum mechanics to those with general
point singularities and walls. The self-adjointness of the supercharge in
relation to the self-adjointness of the Hamiltonian is also discussed.
