The non-convex nature of the power flow equations poses a challenge for solving various power system optimization and control problems. To address these challenges, linear approximations are often employed. However, the accuracy of these linearizations can vary depending on the characteristics of the systems and the operational range. Existing linearizations typically rely on general assumptions that apply to broad classes of systems, which can result in constraint violations. In contrast to these existing approaches, we introduce conservative linear approximations of the power flow equations that intentionally over- or underestimate quantities of interest, aiming to make algorithms more tractable while avoiding constraint violations. Additionally, we introduce rational approximations with linear numerators and denominators. This choice is motivated by the resulting linear inequality constraints, making these approximations well-suited for optimization formulations, while still providing enhanced accuracy compared to linear functions. We enhance the conservativeness and accuracy of our approximations through an iterative sampling method. To further develop our approach, we establish an importance sampling method for constructing conservative linear approximations. This method's objective is to efficiently select the most informative samples. It does so by drawing samples from a relatively low-dimensional subspace exhibiting high curvature. This approach allows us to obtain highly accurate linear approximations with significantly fewer samples than random selection. Furthermore, we examine applications of conservative linear approximations in an optimal sensor placement problem that we formulate as a bilevel program. Our goals are to place a minimal number of sensors, avoid false sensor alarms, and ensure that sensors will detect any voltage violations. To make the bilevel problem tractable, we replace the nonlinear power flow equations with conservative linear approximations and apply various problem reformulations to significantly improve computational tractability while simultaneously ensuring an appropriate placement of sensors.