The Shrimps came closest to breaking the deadlock after laying siege to the visitors' goal in the final stages of the game, but they were denied by two superb saves from Shots keeper Nikki Bull.
Lee Elam had the best chance of the game five minutes from time when free at the far post. Unfortunately he drilled his shot into the floor and Bull was able to scamper across his goal and make an excellent save.
The impressive Bull also denied the Shrimps two minutes earlier when Wayne Curtis let fly with a right-foot shot that was superbly saved by the keeper.
Although the Shrimps looked the most likely to take the points it was Aldershot who felt they had stolen the victory in injury time. A right-wing corner was flicked on and the on-loan Jake Robinson volleyed home at the far post only to see his celebrations cut short by an offside flag.
The game may have ended frantically, but for the first hour chances were at a premium as both sides looked scared to lose ground on the other.
Bull did well to save at the feet of Michael Twiss and block a Garry Thompson effort with his legs while at the other end new Aldershot signing Gary Holloway should have done better than head straight at Ryan Robinson when he had a free header six yards out.
The visitors' best chance fell on 49 minutes when Tim Sills sneaked in front of his marker at a free-kick but volleyed over the bar from close range.
After that scare Morecambe went on the offensive with Curtis and Twiss producing a series of dangerous crosses that were crying out for a touch as Aldershot hung on desperately for what could be a vital point for Terry Brown's side.