  In this paper we explore the possibility of explaining the muon $g-2$ anomaly
in various types of supersymmetric extensions of the Standard Model. In
particular, we investigate and compare the phenomenological constraints in the
MSSM with stable neutralino and the other types of scenarios where the
neutralino is unstable. For the latter case we study the Gauge Mediated SUSY
Breaking (GMSB) scenario with very light gravitino and the $UDD$-type R-Parity
Violating (RPV) scenario. In the MSSM with stable neutralino, the parameter
region favoured by the $(g-2)_\mu$ is strongly constrained by the neutralino
relic abundance and the dark matter direct detection experiments, as well as by
the LHC searches in the lepton plus missing transverse energy channel. On the
other hand, the scenarios without stable neutralino are free from the dark
matter constraints, while the LHC constraints depend strongly on the decay of
the neutralino. We find that in GMSB the entire parameter region favoured by
the muon $g-2$ is already excluded if the Next Lightest SUSY Particle (NLSP) is
the neutralino, while some regions are still allowed if the NLSP is stau. In
the RPV scenario, the LHC constraints are much weaker than the other scenarios
and a wide region of the parameter space is still open for the muon $g-2$.
