<p>What factors determine which approach is more appropriate?</p>

Not nearly enough information here. It depends if your language even supports the construct "Thing.something" or equivalent (ie. it's an OO language). If so, it's far more appropriate because that's the OO paradigm (members should be associated with the object they act on). In a procedural style, of course, DoSomethingtoThing() is your only choice... or ThingDoSomething()