It's difficult to get hard numbers. For comparison, I found this summary of adverse effects involved with Hepatitis vaccines that suggests that serious (life-threatening) adverse effects are very rare, less than 1 in 60,000 (less than 0.002%). I would consider that "generally safe". Certainly I didn't hesitate to get my hepatitis vaccine shots when I needed them.

(Note that mild symptoms are absolutely normal and nothing to worry about -- that's just your immune system at work.)

Keep in mind that there are at least three major Covid vaccines in use in the West, and it is absolutely wrong to count them all together. But having said that, I'm going to do it anyway, because it's 2am here and fuck it.

Estonia found five serious adverse effects after 39,505 doses of two different vaccines were administered. That's about 0.01%. Nobody died out of the five. In comparison, the fatality rate for Covid is about ten times that, plus many more left with long-term damage. So on the very limited information I have now, I would say the vaccines are looking better than Covid as far as side-effects go.

(One in a ten thousand chance of a serious, but probably not fatal, adverse event, versus one in a thousand chance of dying, and about one in fifty chance of serious, long-term debilitating damage to lungs, kidneys, heart, nerves and other organs.)

The Smithsonian is reporting that up to 80% of people will have mild side-effects, but doesn't comment on the risks of serious side-effects.

The medical advice regarding the A-Z vaccine is constantly changing as new data comes in, but I think that people under 50 should probably avoid it. It seems to be broadly safe for the elderly. As for the others, Pfizer and Moderna, and the rest, a lot of it depends on where you are, how much community transmission of Covid there is, the availability of treatments (and the willingness of doctors to prescribe them), and whether or not you and your community is willing to use other preventative measures like lock-downs, stay-at-home orders, social distancing and masks to control the spread of the virus.

I am immune-compromised, but I live in a country with very little community transmission thanks to pretty good compliance with masks and lock-downs. So I'm lucky enough to have the luxury to take a "wait and see" attitude to the vaccines. But if my government is stupid enough to open the international borders, and we get an influx of Covid cases, I would have to seriously consider getting vaccinated as the "least worst" alternative.