  The existence of a fundamental length (or fundamental time) has been
conjectured in many contexts. However, the "stability of physical theories
principle" seems to be the one that provides, through the tools of algebraic
deformation theory, an unambiguous derivation of the stable structures that
Nature might have chosen for its algebraic framework. It is well-known that $c$
and $\hbar $ are the deformation parameters that stabilize the Galilean and the
Poisson algebra. When the stability principle is applied to the
Poincar\'{e}-Heisenberg algebra, two deformation parameters emerge which define
two length (or time) scales. In addition there are, for each of them, a plus or
minus sign possibility in the relevant commutators. One of the deformation
length scales, related to non-commutativity of momenta, is probably related to
the Planck length scale but the other might be much larger. In this paper this
is used as a working hypothesis to look for physical effects that might settle
this question. Phase-space modifications, deviations from $c$ in speed
measurements of massless wave packets, resonances, interference, electron spin
resonance and non-commutative QED are considered.
