An apparatus for checking dimensions and/or shape includes one or more feelers (7,8;9;67,68), three for preference, adapted to touch the surface of a mechanical part (2;61) to be checked, a transducer and/or display device (33;78) with a movable part (34;77), and a mechanical transmission assembly (20) to transmit to the movable part of the transducer and/or display device the displacements of the feeler/s. The mechanical transmission assembly includes a shaft (22) defining an axis (A) and a guiding element (30) adapted to house a guide portion (25) of the shaft. The shaft includes a resiliently deformable portion (26), that is preferably a reduced diameter portion having axial symmetry, in an intermediate position between the guide portion and an abutment portion (27).