One of the nice things about art is that there are no rules. Just opinions that people try to sell as rules. I both agree and disagree on this one - there are no rules, but there are some guidelines like the ones Phil posted that certainly make a portrait look better. For example, cropping at the wrist, knee, or ankle joints may not break any rules, but it's not going to make for a pretty portrait.
I agree. I think it's a good guideline
to follow "rules" within a range of scale. The range will encompass 2 or 3 different zones of wildness, risk and craziness, green, yellow and red. The further out you go on the scale, the more risky (and perhaps more fun) your shots become. In other words, do "establishment (by the rules)" shots first and then branch out and try more risky and unorthodox things to get unique and inventive results. That way, you can please everyone, the client, your own art needs and whatever may be in between. You also learn more this way! So shoot a few good basic shots, then start getting tighter or wider, tilt the camera, have the subject jump, get wacky, etc.
One example, I tell small groups to get ready, be silly, etc and fire off a few shots before they are ready and this often makes them laugh and I fire off more shots. Which makes them laugh more. Then I get serious, they stiffen up, we shoot more (insurance shots) and then I ask them to loosen up and I shoot more. Funny faces! Have fun! When I'm done, my goal is to have about 15+ shots, 3 or 4 of which might be worth keeping and hopefully will be different from one another.
Another thing to remember with portraits, esp one on one shoots is to put your subject at ease. Don't start shooting until they (and you) are relaxed and comfortable. This may take 30 seconds or 30 minutes. But the great photographers out there that make legendary photos of big celebrities, etc do exactly that. They make it a point to get to know and become comfortable with the subject before they start shooting, even if all they are looking for is one good shot to have at the end.