This article addresses the critical issue of testing static approximate factor models against unspecified alternatives by proposing an adaptive residual-based test. Our methodology is versatile, encompassing alternatives such as factor models with nonlinear factor structures, conditional factor models, and factor models with structural breaks. We first establish the asymptotic properties of a so-called fixed bandwidth-based test, showing its asymptotic normality and power against local alternatives. To enhance the size and power properties while addressing the practical limitations of the fixed bandwidth approach, we propose an adaptive test and establish its asymptotic size correctness and consistency. Simulation studies demonstrate its superior size and power. We further illustrate the practical relevance of our test by applying it to three empirical studies concerning real and financial variables in the global economy.