Should an undergraduate summer research project be put under "employment history"? I am currently filling out an application for gradute study, which includes a section on employment history. 

During my time as an undergraduate, I carried out two summer research projects in pure maths. One of them was based at my home institution, for which I obtained a monthly stipend from an outside organisation (a research bursary). For the second project, I was on the payroll of the university at which the project was based. 

The application asks for details of "work experience, internships or volunteer work". Do either of these count?
Yes they count.


  "work experience, internships or volunteer work"


is clearly more broad then just regular employment. Just make clear what type of position each role was. If there is some other section that allows you to report other types of research experience you could be free to classify these either way, it wouldn't affect much. The people reading your application just what to know what you've done.