  We demonstrate the formation of stable spatially-ordered states in a {\it
uniform} and also {\it trapped} quasi-two-dimensional (quasi-2D) Rashba or
Dresselhaus spin-orbit (SO) coupled pseudo spin-1/2 Bose-Einstein condensate
using the mean-field Gross-Pitaevskii equation. For weak SO coupling, one can
have a circularly-symmetric $(0,+1)$- or $(0,-1)$-type multi-ring state with
intrinsic vorticity, for Rashba or Dresselhaus SO coupling, respectively, where
the numbers in the parentheses denote the net angular momentum projection in
the two components, in addition to a circularly-asymmetric degenerate state
with zero net angular momentum projection. For intermediate SO couplings, in
addition to the above two types, one can also have states with stripe pattern
in component densities with no periodic modulation in total density. The stripe
state continues to exist for large SO coupling. In addition, a new
spatially-periodic state appears in the uniform system: a superlattice state,
possessing some properties of a supersolid, with a square-lattice pattern in
component densities and also in total density. In a trapped system the
superlattice state is slightly different with multi-ring pattern in component
density and a square-lattice pattern in total density. For an equal mixture of
Rashba and Dresselhaus SO couplings, in both uniform and trapped systems, only
stripe states are found for all strengths of SO couplings. In a uniform system
all these states are quasi-2D solitonic states.
