How effective and safe are measles vaccines?

A couple good meta-questions - but perhaps only to ask yourself, silently - when someone is talking up the real or imagined dangers of vaccines:

- Has this person talked to a few extremely old folks, who remember how things were before the vast majority of today's vaccines were first developed? (Hint: Compared to the grim realities of the diseases they mostly prevent, vaccine side-effects are utopia.)

- When the topic is not as hot and trending as vaccine side-effects, does this person seem to care about long-odds risks? Say, avoiding all alcohol and cured meats because of the cancer risks? Testing their smoke detectors every month? Avoiding any body of water large enough to drown in? Keeping their CPR and First Aid certification up-to-date?