Isosurfaces are the key entities in visualization, and recent research has focused on the visualization of uncertain data. We study positional uncertainty in isosurfaces when data uncertainty is modeled with parametric and nonparametric density models. Noise in sampled data leads to topological and geometric variations in the extracted isosurface. We propose probabilistic techniques that reduce topological uncertainty in isosurfaces and resolve ambiguities in identifying topological configurations. We also characterize, in closed form, a random variable modeling the geometric uncertainty in isosurface extraction. Closed-form characterization of geometric uncertainty allows analytic computation of expected value of level-crossing location, as well as the variance. While former quantity can be used for constructing a stable isosurface for uncertain data, the latter can be used for visualizing the spatial uncertainty in the extracted isosurface. We show computational advantage of proposed analytic approach over expensive Monte-Carlo sampling approach for quantifying geometric uncertainty. The advantages of nonparametric statistical models over parametric noise models for uncertainty characterization are evident from our experiments on ensemble datasets and uncertain scalar fields. We also propose a novel approach that leverages nonlocal statistics for accurate characterization of underlying uncertainties in the nonparametric framework.