When to consider yielding vs. rupture as a mode of failure?

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Are there any general rules of thumb for which to choose? I have a lot of ambigous HW problems which don't specify, they just list a safety factor. For example, if I have a mass hanging on a metal wire with a specified safety factor, then I mean it will buckle at the yield strengh but only rupture at the ultimate strengh. Is it really just discretionary?
 
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In general yielding is not the same as failure. Some types of engineering components made from ductile materials are intentionally stressed beyond the yield point. Brittle materials often fail in fracture initiated by internal micro-cracks, poor surface finish, etc, before they yield.

For homework questions, you probably want to apply the safety factor to the ultimate stress. In a real life design situation, the design criteria should not be ambiguous, and for safety-crifical items they will often be taken from national or international standards..

For example, if I have a mass hanging on a metal wire with a specified safety factor, then I mean it will buckle at the yield strengh
Two comments on that.
1. If the wire is in tension, it won't buckle at all, in the standard engineering meaning of the word buckle.
2. Buckling loads depend on the geometry of the component and Young's modulus, not on the strength of the material. The stress level for buckling can be orders of magnitude smaller than the yield or ultimate stress.
 
it won't buckle at the yield strength, the yield strength denotes the onset of plastic deformation. failure only occurs at the UTS. In that sense I don't think yielding can ever be considered a failure mode. It can lead to things like fatigue which is a failure mode due to cyclic loading and also creep failure under constant loading with a constant strain rate.
Yielding is when something no longer deforms elastically, but deforms plastically which is irreversible and causes a permanent strain in the material.