Which stars have planets at all?

Not all stars have planets. There are three classes of stars which are unlikely to have planets:

• Members of close binary or multiple systems. In this case, the stars would disturb each other's planetary orbits. As a rule of thumb, no orbits with radii of more than 1/3 of the minimum distance between the two stars are stable. For planets bearing life, the orbit radius must be even smaller compared to the minimum star distance to allow stable climatic condotions. Alpha Centauri is probably the closest where something like Earth could exist. However, even if the stars are far enough apart to allow for stable orbits of planets, their tidal forces might preclude the formation of planets, just like the tidal forces of Jupiter might be the reason why the asteroids between Mars and Jupiter did not combine into a compact planet.
• Fast-rotating stars. These have an angular momentum similar to that of the entire solar system. The most probable reason why they have retained such a high angular momentum is simply that they have no planets to share their angular momentum with, except perhaps some small far-out terrestrials. Most O and B stars and many A stars fall into this category. These are the big ones who are too short-lived to let higher life-forms on their planets evolve anyway. The smaller F, G, K and M main-sequence stars rotate slowly, as our sun does, and probably have planets.
• Population II stars. When they were formed, elements heavier than helium were very scarce. Thus, they don't have planets, except perhaps hydrogen-helium giants which would be, because of the absence of heavier elements, colourless and sterile.

There is a possible explanation why large stars seem to be planetless. Within each solar system, there is an inner limit within which planets cannot form because it is simply too hot. Above about 1000 or 2000 degrees C, not even metals and silicates condense any more. In our solar system, this limit is quite close to the sun, far within Mercury's orbit. But a huge O star is so hot that even the outer fringes of its protoplanetary disk won't condense.

This leaves us with the medium to small population I stars. Most of these stars are still on the main sequence (the sun has a main-sequence lifetime of 10000 million years, thus a red giant of solar mass is at least 10000 million years old, and that was the time when the last population II stars were born).