  The problem of adaptive multivariate function estimation in the single-index
regression model with random design and weak assumptions on the noise is
investigated. A novel estimation procedure that adapts simultaneously to the
unknown index vector and the smoothness of the link function by selecting from
a family of specific kernel estimators is proposed. We establish a pointwise
oracle inequality which, in its turn, is used to judge the quality of
estimating the entire function (``global'' oracle inequality). Both the results
are applied to the problems of pointwise and global adaptive estimation over a
collection of H\"{o}lder and Nikol'skii functional classes, respectively.
