I know neutrinos cant oscillate in the SM due to lack of mass but why are other processes forbidden? Ie. $\mu\rightarrow e\gamma$
The standard model has conservation laws which are derived from a large number of obsrvations and experiments:
In developing the standard model for particles, certain types of interactions and decays are observed to be common and others seem to be forbidden. The study of interactions has led to a number of conservation laws which govern them. These conservation laws are in addition to the classical conservation laws such as conservation of energy, charge, etc., which still apply in the realm of particle interactions. Strong overall conservation laws are the conservation of baryon number and the conservation of lepton number. Specific quantum numbers have been assigned to the different fundamental particles, and other conservation laws are associated with those quantum numbers.
Different lepton numbers are assigned to the three generations of leptons, tau, mu, e and this has the consequence that conserving them in the decay means that in muon decay a muon neutrino has to be there to conserve muon lepton number. That is the true state of the standard model.
but why are other processes forbidden?
It is a consequence of imposing the observational fact that lepton number is conserved
Neutrino oscillations are beyond the standard model because they have masses and the reaction μ--> e γ could happen with very small probability through a neutrino oscillation diagram. See my answer here and the limits in this experiment.