Yield stress fluids do not deform unless a stress threshold is exceeded, but are otherwise viscous. The presence of the yield stress significantly alters the stability properties of these flows. In the first place, many static flows which would be mechanically unstable for simpler Newtonian fluids can now be observed. These flows are found when the ratio of yield stress to driving stresses exceeds a critical value, say Y Y_c. Secondly, in the case that Y Y_c the static flows appear to be energy stable (and globally energy stable in some cases). The critical value Y_c is defined by an optimization problem that may be solved approximately by a number of methods that we outline. An interesting observation is that, when YY_c, as the transient velocity decays to zero the slowest decaying components of the solution resemble the minimizer that defines Y_c. Our intuitive interpretation of this is as a nonlinear eigenvalue problem. To illustrate the behaviour of these systems, we present examples from applications with particle motion and with buoyancy driven motion. Joint work with E. Chaparian, I. Karimfazli and A. Wachs.