The few times I have tried to capture a rainbow the results have been washed out. Although I have found a polarizing filter to be very useful for sky photography, I'm guessing that it would wipe a rainbow out, since it is basically a scattered light phenomenon. What are the aperture/shutter trade-offs?
Any rules of thumb would be helpful, but I'm especially looking for the gotchas to avoid, since time tends to be limited when a rainbow makes an appearance.
I don't think there's a lot you can do photographically. I think the quality of the photo will come down to how strong the rainbow is, and how you process the image (either by the choice of picture style/JPEG saturation level in camera, or Raw processing).
It takes a combination of factors which all have to come together to form a strong rainbow — you need strong direct sunlight, but the sun also has to be low in the sky. If it's above 42 degrees then the rainbow will fall below the horizon. Generally, when the sun is lower in the sky, it's not as strong. Midday in the winter, or morning/late afternoon in summer will give you the best strength to angle ratio, though it depends on your latitude. You also need an abrupt boundary between clear sky and rain, such as you get with a heavy thunderstorm. These also tend to bring wind, which disperses the cloud — making the rainbow shortlived. This is why really good rainbows happen rarely: they require many competing factors.
Aperture/shutter speed aren't going to make a huge difference, provided your shutter is fast enough to avoid camera shake. You're going to need a fairly wide lens to capture a rainbow, so depth of field is not going to be a problem. I'd opt for a usually-optimum aperture of f/5.6 (or maybe f/8). The main arc of a rainbow is always 42 degrees from the centre, so you need an 84° FOV to capture an entire rainbow. This corresponds to at least 20mm lens (full frame) or 12.5mm lens on a crop body. Rainbow light is strongly polarized, so take off the filter (as mmr states).
For post processing, a contrast/saturation boost will help bring out the rainbow colours and give the image some punch.
I would definitely remove the polarizer, and also try to set your white balance properly. Because the rainbow is the entire spectrum of light, if your white balance is off, the rest of the image can look strange, or the rainbow can look strange, when you correct. Also, other forms of color correction and the like (saturation adjustment, etc) will feel the same effect; essentially, since the rainbow contains all colors, emphasizing one color over the others results in odd final images because then the rainbow will be unbalanced.
Not much to tell about shooting rainbows, other than what's already written in the other questions, but most of a good rainbow photo is good composition a bit of luck of being on the right place. Still, you could try to slightly under expose your photo to boost the rainbow's colors.
