Vandals damaged high school property and an old water works in separate attacks last weekend, police said.
A bench at Brae High School was damaged sometime between 5pm Friday and 8am Monday.
Vandals also struck the old water treatment works near the A970 Black Gaet junction with the B9073 near Scalloway at sometime over the same weekend.
Officers at Lerwick police station are asking for anyone with information to get in touch.
PC Andrew MacKenzie is also appealing for information regarding a theft, in which garden furniture was stolen from a garden in the Sandveien area of Lerwick.
It happened sometime between 6pm on Saturday and 10am the following morning.