Elastic waveguiding structures are investigated for two-dimensional guides constructed of periodi- cally alternating media thereby forming a striped waveguide. The guide has finite width with either homogeneous traction-free or clamped boundary conditions on the guide walls, or a combination of these. The band spectrum and associated Floquet-Bloch eigensolutions for these in-plane elastic waveguides are identified. Several features of this guiding structure emerge, and are of interest, in some cases a total stop band at zero frequency is identified providing space for low frequency lo- calised modes: such modes also appear when we create defects in the structured waveguide. The dispersion curves often have maxima and minima of the spectral edges far from the edges of the Brillouin zone and these are related to slow sound or standing waves within the structure. Numerical and asymptotic techniques are developed and discussed, the latter are based on weak contrast, and weak geometric changes, or utilize jump conditions in limits where one medium is thin relative to the other. There are technological applications that could utilize this theory and we demonstrate that subwavelength imaging is possible in some regimes.