  We develop a general framework for the description of instabilities on soap
films using the Bj\"orling representation of minimal surfaces. The construction
is naturally geometric and the instability has the interpretation as being
specified by its amplitude and transverse gradient along any curve lying in the
minimal surface. When the amplitude vanishes, the curve forms part of the
boundary to a critically stable domain, while when the gradient vanishes the
Jacobi field is maximal along the curve. In the latter case, we show that the
Jacobi field is maximally localised if its amplitude is taken to be the lowest
eigenfunction of a one-dimensional Schr\"odinger operator. We present examples
for the helicoid, catenoid, circular helicoids and planar Enneper minimal
surfaces, and emphasise that the geometric nature of the Bj\"orling
representation allows direct connection with instabilities observed in soap
films.
