In this paper, we investigate general single-index models (SIMs) in high dimensions. Based on U-statistics, we propose two types of robust estimators for the recovery of model parameters, which can be viewed as generalizations of several existing algorithms for one-bit compressed sensing (1-bit CS). With minimal assumption on noise, the statistical guarantees are established for the generalized estimators under suitable conditions, which allow general structures of underlying parameter. Moreover, the proposed estimator is novelly instantiated for SIMs with monotone transfer function, and the obtained estimator can better leverage the monotonicity. Experimental results are provided to support our theoretical analyses.