This paper addresses the problem of selection and discovery of a consistent availability monitoring overlay for computer hosts in a large-scale distributed application, where hosts may be selfish or colluding. We motivate six significant goals for the problem - consistency, verifiability, and randomness, in selecting the availability monitors of nodes, as well as discoverability, load-balancing, and scalability in finding these monitors. We then present a new system, called AVMON, that is the first to satisfy these six requirements. The core algorithmic contribution of this paper is a protocol for discovering the availability monitoring overlay in a scalable and efficient manner, given any arbitrary monitor selection scheme that is consistent and verifiable. We mathematically analyze the performance of AVMON's discovery protocols, and derive an optimal variant that minimizes memory, bandwidth, computation, and discovery time of monitors. Our experimental evaluations of AVMON use three types of availability traces - synthetic, from PlanetLab, and from a peer-to-peer system (Overnet) - and demonstrate that AVMON works well in a variety of distributed systems.