We develop both stable and stabilized methods for imposing Dirichlet constraints on embedded, three-dimensional surfaces in finite elements. The stable method makes use of the vital vertex algorithm to develop a stable space for the Lagrange multipliers together with a novel discontinuous set of basis functions for the multiplier field. The stabilized method, on the other hand, follows a Nitsche type variational approach for three-dimensional surfaces. Algorithmic and implementational details of both methods are provided. Several three-dimensional benchmark problems are studied to compare and contrast the accuracy of the two approaches. The results indicate that both methods yield optimal rates of convergence in various quantities of interest, with the primary differences being in the surface flux. The utility of the domain integral for extracting accurate surface fluxes is demonstrated for both techniques. Copyright © 2011 John Wiley & Sons, Ltd.