Parcel bombs sent to Celtic manager Neil Lennon, a lawyer and a politician were "designed to cause real harm to the person who opened them", police said.
Two suspect packages were addressed to Lennon and one to Paul McBride QC and former Labour MSP Trish Godman.
Strathclyde Police Detective Superintendent John Mitchell said sending the packages was a "despicable and cowardly act".
Strathclyde Police confirmed they launched a major investigation after two "suspect packages" were intercepted at Royal Mail sorting offices last month, both addressed to the Celtic manager.
The first one was found in Saltcoats, Ayrshire, on March 4 and the second was intercepted on March 26 in Kirkintilloch, East Dunbartonshire.
Two days later, on March 28, a parcel for Ms Godman, the former deputy presiding officer of the Scottish Parliament, was delivered at her constituency office.
The most recent package was intercepted last Friday at a post-box on Montgomerie Terrace in Kilwinning, Ayrshire, and was addressed to Mr McBride, who represented Lennon at Hampden during his recent dispute with the Scottish Football Association (SFA).