Mechanical separators and screening machines, and methods for flexible sieve mat screening and flexible mat conveying are disclosed. In an example configuration, a flexible mat screening apparatus is provided with geometrically optimized guiding edge seals at lateral sides. One preferred configuration includes an assembly wherein a sieve mat has upwardly curved lateral sides forming a non-vertical, gradually curved shape. In another configuration, a movable support section is supported on a main frame section via a plurality of shear blocks, each arranged with its compression axis disposed horizontally. In yet another configuration, the movable support section is further connected to the main frame section via vertical stabilizers or leaf springs, the vertical stabilizers permitting longitudinal movement between the movable support section and the main frame section, but inhibiting vertical and/or lateral movement therebetween. Yet another configuration includes a boltless mat section attachment assembly.