The stalemate means the U's have now gone 19 league matches without winning.
Colchester's Chris Porter wastefully slotted wide when one-on-one with Town goalkeeper Jayson Leutwiler before Shrewsbury's Abu Ogogo rattled a post.
Town striker Andy Mangan dragged a low second-half shot wide when in the clear, before Leutwiler pushed away substitute Elliot Lee's shot.
Colchester, who now visit Bradford City on Tuesday night, stay bottom, 10 points adrift of 21st-placed Shrewsbury.
Micky Mellon's side, beaten in the FA Cup by Manchester United on Monday night, remain in the drop zone on goal difference, ahead of their midweek home game with Rochdale.
Shrewsbury Town manager Micky Mellon told BBC Radio Shropshire:
"It was one of those games where both sides were desperate for the points. We had a bit of a hangover from Monday, but I can see positives.
"You have to respect a point on the road. It's a clean sheet away from home. Sometimes you have to understand the circumstances surrounding the game and respect that point.
"It's another away point but we move on to the next game on Tuesday. Rochdale is big. We will approach it in such a way that we will be determined to pick up the three points."
Colchester United set a new club record for league games without a win after a goalless draw with Shrewsbury Town.