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Through rigorous analysis of all possible oscillatory guided modes propagating in the cylindrical and slab dielectric waveguides with metamaterial cladding, we demonstrate that slow or even stopped light can propagate in the waveguides with specially selected parameters of the metamaterial. We propose the design of linearly tapered waveguides which could lead the propagating light to a complete standstill and validate the structure by full-wave simulation based on finite-difference time-domain (FDTD) method. For practical implementation, we also discuss the effect of the material loss on the slow light modes and propose a method by incorporating gain in core material to compensate the loss in the cladding material and regain a complete standstill of the light.