This is a great thread. I actually do a fair bit of portraiture for my job (albeit mostly the canine variety) and I had to figure this out as I went. There are a few questions to answer before deciding on the appropriate aperture for a portrait:
1) What focal length lens are you going to use? This deals mostly with the type of portrait. If you're wanting to shoot close up and personal and just frame the head and tops of the shoulders, I find a 50-75mm lens gives a decent working distance while keeping things fairly personal. You can of course, move further back and choose a longer focal length, maybe 100-135mm (zooms are a great way of framing these types of shots).
2) Desired of depth of field? This is where the creative control comes in. For me, I like to have the majority of the head "in focus" that is, without major blurring of the major facial features. For animals, especially dogs, you really want to have the nose in focus as well as the ears and with larger dogs, you can be talking about 10-12 inches for this span.
Once you decide these two basic criteria, a quick check of your depth of field calculator, I really like
The Online Depth of Field Calculator. If you have a lens with DOF markings you can do this on the fly until you get the hang of the differences.
I can simplify this based on my field experiences to say that when I shoot a medium-sized animal, I try to use a 50mm lens on my APS-C sensor camera making it an equivalent 75mm lens. My working distance is usually in the 6-8 foot range and I shoot at around f/8 for a sufficient DOF. This provides a very normal-looking perspective. If I want a blurrier, less distracting background, I can always reposition the model so they are further away from the objects in the background. I find I prefer this over opening up the aperture as I begin to lose the desired DOF.
For a more creative portrait look, however, I'll switch to a longer focal length (75-90mm) and move back to around 10-12 feet and shoot at the same aperture (f/8), you have a bit more leeway regarding background objects as the longer FL will blur them more easily.
I rarely shoot normal portraits at less than 50mm or longer than 135mm (actual settings on the lenses, not crop-factor numbers) as I find that <50mm leads to a bit too much distortion and >135mm requires too fast of a shutter speed to avoid camera shake. This is a very conservative approach to portraiture, but I'm not going for creative composition but instead a very straightforward setup that allows me to get very predictable (some might rightfully say boring) results. It's at least a good starting point before you jump into more unique options.