There is a third option on many cameras that is worth knowing about because it is often the best default: electronic first curtain.
It starts the exposure electronically and ends it with the mechanical shutter. That removes the vibration from the shutter opening, which is the part that causes shutter shock - while keeping a mechanical closing, which avoids most rolling shutter effects.
The catch is that at very fast shutter speeds with a wide aperture it can produce uneven exposure or oddly shaped out-of-focus highlights, because the electronic start and mechanical finish do not travel identically.
So a reasonable set of defaults:
- Electronic first curtain for most work.
- Full mechanical for very fast speeds with fast lenses, and whenever you use flash.
- Fully electronic for silence, and for static subjects where vibration matters.
- Avoid fully electronic for fast motion and under artificial lighting.