Shenton, 30, joins the club for his third spell after departing Premier Division rivals Buxton.
The goalkeeper first joined Celtic in 2016 from Ashton United, and returned to Bower Fold on loan last season.
Grant made a club record 279 appearance for Ramsbottom United after joining the club from FC United of Manchester. He has also played for Trafford and Chester, where he kept 16 clean sheets in 42 games before most recently joining Buxton.
Burton, 33, joins the club after a brief spell with Bamber Bridge.
The experienced midfielder had spent two years at Chester where he made 41 appearances for the club under Anthony Johnson and Bernard Morley as well as becoming club captain, and was previously with Salford City.
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