The 30-year-old forward ends a four-year stay with Aberdeen to link up with the Shrews as they look to bounce back from relegation at the first attempt.
Relegation saw the club appoint former Fleetwood manager Micky Mellon as their new boss.
"Scott is a very good player, a clever striker, who I believe will fit in perfectly," said Mellon.
"He makes good runs and has good movement. He's a striker, but can play in a number of variations. He can play in the number 10 role, he can play in a front two or up front on his own."
Another relegated club, Tranmere Rovers, have also strengthened their squad with the signing of Michael Ihiekwe. The 21-year-old defender has signed a two-year deal after being freed by Wolverhampton Wanderers.