In this paper, the magnetohydrodynamic flow of an electrically conducting, viscoelastic fluid past a shrinking sheet is investigated. The main concern is to analytically examine the structure of the solutions and determine the thresholds beyond which the pure exponential type of solution ceases to exist or multiple solutions exist. Closed-form solutions for the boundary-layer equations, which are exact solutions of the full governing Navier-Stokes equations of the flow, are presented for two classes of viscoelastic fluid, namely, the second-grade and Walter liquid B fluids. The flowfield is affected by the presence of physical parameters, such as viscoelasticity, magnetic, and suction/injection parameters, whereas the temperature field is additionally affected by thermal radiation, heat source/sink, Prandtl, and Eckert numbers. The regions of existence or nonexistence of unique/multiple solutions sketched by the combination of these parameters are initially worked out by providing critical values, and then velocity/temperature profiles and skin friction coefficient/Nusselt numbers are examined and discussed. Heat transfer analysis is also carried out for two general types of boundary heating processes, either by a prescribed quadratic power-law surface temperature or by a prescribed quadratic power-law surface heat flux. The shrinking sheet magnetohydrodynamic flow of viscoelastic fluid shows richer nonlinear behavior in the solution domain with multiple solutions.