It has been understood for some time that elastic energy can cause surface roughening during a solid surface motion. This instability has recently led to a novel experimental technique to determine stress state on the surface of a solid by measuring the surface profile before and after etching [1]. Along a separate line of investigation, Aziz and co-workers has recently described a different kind of instability, also driven by stress [2]. Their experiments showed that the activation energy of the surface mobility depends linearly on the stress state, and this dependence can cause surface instability. The two kinds of instabilities have very different characteristics. In this paper, we describe a linear stability analysis of a three dimensional interface evolving under stress. The interface can be destabilized either by stress-dependent activation energy or by elastic energy. The implications for the stress measurement technique are discussed. It is suggested that the same experimental procedure be used to measure surface energy and activation strains.