The threat of false data injection (FDI) attacks have raised wide interest in the research and development of smart grid security. This paper presents a comparative study on the utilization of supervised learning classifiers to detect direct and stealth FDI attacks in the smart grid. A detailed formulation of the problem for detection with classifiers is first described with proper assumptions and justifications. Three widely used supervised learning (SL) based classifiers are chosen to design corresponding FDI detectors. The performance are tested against false measurement data (direct FDI attack) and false state data (stealth FDI attack) on both balanced and imbalanced cases, with consideration of the influence of FDI resources and magnitudes. Simulations on IEEE 30-bus system have shown that the SL based detectors can effectively detect both direct and stealth FDI attacks, especially for the more severe attacks with large amount or magnitude of compromised measurements.