We consider the problem of selecting an optimal set of sensors to estimate the states of linear dynamical systems. Specifically, the goal is to choose (at design-time) a subset of sensors (satisfying certain budget constraints) from a given set in order to minimize the steady state error covariance produced by a Kalman filter. In this paper, we show that this sensor selection problem is NP-hard, even under the additional assumption that the system is stable. We then provide bounds on the worst-case performance of sensor selection algorithms based on the system dynamics, and show that certain typical objective functions are not submodular or supermodular in general. While this makes it difficult to evaluate the performance of greedy algorithms for sensor selection, we show via simulations that a certain greedy algorithm performs well in practice. We also propose a variant of the greedy algorithm which is based on the Lyapunov equation and show that the corresponding (relaxed) cost function is modular.