Both sides had several near misses. Waide Fairhurst hit the post after 11 minutes and Nathan Elder hit the other Grimsby post soon afterwards.
Visiting goalkeeper Nick Colgan was also at full stretch to punch away Jamie Devitt’s free-kick.
Shrewsbury goalkeeper David Button saved brilliantly from Adam Proudlock, who should have done better from eight yards soon after half-time.
Grimsby substitute Barry Conlon shot wide of an open goal 15 minutes from the end after good work by Jean-Louis Akpa-Akpro.
The Mariners have now gone 13 games without a win.