This paper develops test statistics based on scores for the specification of regression in nonparametric and semiparametric contexts. We study how different types of test statistics focus power on different directions of departure from the null hypothesis. We consider index models as basic examples, and utilize sieves for nonparametric approximation. We examine various goodness-of-fit statistics, including Cramer-von Mises and Kolmogorov-Smirnov forms. For a "box-style" sieve approximation, we establish limiting distributions of these statistics. We develop a bootstrap resampling method for estimating critical values for the test statistics, and illustrate their performance with a Monte Carlo simulation.