The 30-year-old left the Scottish club at the end of last season after seeing his contract come to an end.
"Scott is a very good player and a clever striker who I believe will fit in perfectly to what we are doing," Shrews boss Micky Mellon told the club's official website.
"He's got great experience and I am absolutely delighted to have Scott on board. It's a deal we know that we have done for a few weeks now but with Scott being on holiday we had to wait until now."
Vernon scored nine goals in 34 appearances for Aberdeen last season.