Which of the following is correct? “He is busy at weekends” or “He is busy on weekends”?

Both are acceptable. “At weekends” is commonly used in British English; the Americans, on the other hand, tend to use “on weekends.” Use whichever expression you feel comfortable with.

Examples: I informed my new boss that I do not do any office related work at weekends. *My friends and I are planning to go sailing on the weekend.

Source: ‘Know Your English’ (The Hindu) – October 02, 2006.