With Celtic's 2-0 victory over Hearts today, Celtic haven't conceded a goal in their first 5 matches of the league campaign, something they haven't achieved since the 1906/07 season, when Willie Maley as manager.
In 1906/07, Celtic actually managed 6 consecutive games without conceding. Winning in two weeks away against St. Johnstone will match that record, and if the Celtic defence and Schmeichel can reach seven, away against Ross County, they'll take the all-time record.
Celtic's goalkeeper at that time was Davie Adams. However, two days before the start of the season, while playing in a benefit game at Ibrox for Rangers player Finlay Speedie, Adams suffered a laceration in his hand due colliding with a nail that had been inserted into a goalpost for a five-a-side tournament, and which hadn't been removed. Rangers offered their reserve goalkeeper, Tom Sinclair, and it was he who played in those first six games, not losing a single goal.
Here are the results that led to the current record for most clean sheets to start a season:
Saturday 18th August 1906
Motherwell 0 - 6 Celtic
Saturday 25th August 1906
Celtic 5 - 0 Kilmarnock
Saturday 1st September 1906
Morton 0 - 2 Celtic
Saturday 15th September 1906
Celtic 3 - 0 Hearts
Monday 24th September 1906
Celtic 2 - 0 Third Lanark
Saturday 29th September 1906
Airdrieonians 0 - 2 Celtic
This remarkable run ended a couple of weeks later, when Celtic beat Aberdeen 2-1 on the 13th October 1906.
Sources for goalkeeper info:
Jeepers Keepers: A look at the history of Celtic goalies – Part Two - Celtic FC
Sinclair, Thomas - The Celtic Wiki