Surface integral equations, which are commonly used in electromagnetic simulations, have recently been applied to various plasmonic problems, while there is still no complete agreement on which formulations provide accurate and efficient solutions. In this paper, we present the strong material dependences of the conventional formulations, revealing their contradictory performances for different problems. We further explain the numerical problems in the constructed matrix equations, shedding light on the design of alternative formulations that can be more accurate, efficient, and stable than the existing ones. Based on our observations in the limit cases, we present a new formulation, namely, a modified combined-tangential formulation (MCTF), which provides stable solutions of plasmonic problems in wide ranges of negative permittivity values. The favorable properties of MCTF in comparison to other formulations are demonstrated not only on canonical problems but also on realistic cases involving nanowires.