This paper discusses the properties of flexural waves governed by the biharmonic operator, and propagating in a thin plate pinned at doubly periodic sets of points. The emphases are on the design of dispersion surfaces having the Dirac cone topology, and on the related topic of trapped modes in plates for a finite set (cluster) of pinned points. The Dirac cone topologies we exhibit have at least two cones touching at a point in the reciprocal lattice, augmented by another band passing through the point. We show that these Dirac cones can be steered along symmetry lines in the Brillouin zone by varying the aspect ratio of rectangular lattices of pins, and that, as the cones are moved, the involved band surfaces tilt. We link Dirac points with a parabolic profile in their neighbourhood, and the characteristic of this parabolic profile decides the direction of propagation of the trapped mode in finite clusters.