The apparatus comprises two blocks (10, 12) intended to be associated respectively with the fixed part and with the movable part of a press, to be moved close and moved away mutually. Each block (10, 12) is equipped with a respective blank holder (14, 22). The block (12) contains a blade-carrier body (38) which is movable with the block itself and has a lateral bending blade (44). The blank holder (14) of the other block has a channel (16) in which the end part of the body (38) can be inserted and one side wall of which is shaped like a counterblade (18). The blank holder of the block (12) is in the form of a plate (22) which is slidable in relation to its block in the direction of mutual moving close of the two blocks (10, 12) and is pushed back by resilient means (56) towards the other block to exert a resilient pressure for restraining the piece of sheet (P). The relative arrangement is such that the bending blade (44) does not project in relation to the plate (22) when it is situated in the position of maximum removal fromits block (12).