The aim of this paper is to study the behaviour of Bingham materials near the apex of a corner for different external flows: (a) Poiseuille flow due to an applied external pressure or other axially moving boundaries; (b) creeping plane flow around wedges; (c) three-dimensional flow in the neighbourhood of the corner of the base of a right circular cylinder. For sharp corners (2-alpha > pi) of included angle 2-alpha, an exact expansion of the flow field in the neighbourhood of the corner is obtained; this is arbitrary up to a multiplicative constant. The leading (singular strain rate) term reflects the fluid's local Newtonian behaviour and higher-order terms reflect the effects of the non-linearity due to the Bingham fluid. For acute angles (2-alpha < pi), arguments are given to suggest plug like behaviour. For case (a), local exact solutions are obtained by means of the Legendre transformation for flow induced by an external disturbance without pressure gradient and corrections for the pressure field are also given. These results indicate the existence of a plug region in the neighbourhood of the corner, and an exact formula for the locus of a plug region is obtained, for example: r2(cos2/3 theta + sin2/3 theta)3 = 4A2, for the case 2-alpha = pi/2 in the absence of the pressure gradient. A is a multiplicative constant. Non-existence arguments are given to show that separable solutions are not possible for case (b).