Route instability is an important contributor to data plane unreliability on the Internet and also incurs load on the control plane of routers. In this paper, we study how route selection schemes can avoid these changes in routes. Modifying route selection implies a tradeoff between stability, deviation from operators' preferred routes, and availability of routes. We develop algorithms to lower-bound the feasible points in these tradeoff spaces. We also propose a new approach, Stable Route Selection (SRS), which uses flexibility in route selection to improve stability without sacrificing availability and with a controlled amount of deviation. Through large-scale simulation, a software-router implementation, and an emulation with real-world BGP update feeds, we demonstrate that SRS is a promising approach to safely stabilize route selection.