The fractional stable motion is a prototypical stochastic process exhibiting both heavy tails and long-range dependence, parameterized via a stability index $α$ and a Hurst exponent $H$. We consider a nonstationary extension where the Hurst exponent is a function of time, and potentially random. The construction admits the standard linear fractional stable motion as tangent process, and we exactly determine its local Hölder exponent in terms of the pointwise values of the Hurst function. This is in contrast to other definitions of multifractional processes, where the Hurst function needs to have additional regularity in time.