Here are a few things I use all the time, so I decided to start writing them down instead of always re-deriving them on the fly.
Some were checked with Sage. I tend to make silly mistakes when I think that "it's just manipulations now", so I like the check.
- Multiplication of two gaussians (includes all normalisations, unlike other derivations online which stop at "here is a Normal distribution")
- Convolution of two Gaussians
- Bayesian odds ratio of multinomials I don't know of any piece of software which can check this one, so you'll have to take it on faith.