What is the best way to clean the black grit out of crevices of linoleum flooring?

There can be a lot of ways out to do that. If the Paint has dried hard materials such as Gaffer tape can be used to take it off. Using paste of salt and rubbing alcohol really helps remove older stains. Let the stain soak in rubbing alcohol and then cover the stain in salt and add more rubbing alcohol. Use your fingers to scrub. Never use any sand paper on that as that may lead to loss of smooth finish.