Although nothing was stolen, damage was caused to the doors of the shed and the alarm was activated.
Police believe it may be linked to a similar break-in at Newbattle Golf Club, in Midlothian, on Saturday night where a shed was also targeted and a number of power tools stolen.
Police are asking anyone who has information to come forward.
Sergeant Scott Nicolson said: “As a matter of course we have advised all other golf clubs in the area, to review their security arrangements and we are keen to speak to anyone who noticed any suspicious activity at, or around, Musselburgh Golf Club on Sunday night to come forward immediately.”
Those with information can contact police on 101 or, alternatively anonymously on Crimestoppers on 0800 555 111.