How to know what exercises are safe for people with a bulging disc?

Exercises for disk: Core muscle exercises in general. Start with flexibility first and add strengthening once limber.

Any: Pretty much any - except for high impact sports maybe. Jogging may be less ideal due to the repetitive pressure on the disc. Golfing may be less ideal due to the rotational acceleration putting additional stress on the disc. Isometric back strengthening is best, it provides more muscle power which will keep the biomechanical forces off the discs. (rowing, bicycling, swimming, yoga).