This paper considers the problem of automatic fault diagnosis for transmission lines in large scale power networks. Since faults in transmission lines threatens stability of the entire power network, fast and reliable fault diagnosis is an important problem in transmission line protection. This work is the first paper exploiting sparse signal recovery for the fault-diagnosis problem in power networks with nonlinear swing-type dynamics. It presents a novel and scalable technique to detect, isolate and identify transmission faults using a relatively small number of observations by exploiting the sparse nature of the faults. Buses in power networks are typically described by second-order nonlinear swing equations. Based on this description, the problem of fault diagnosis for transmission lines is formulated as a compressive sensing or sparse signal recovery problem, which is then solved using a sparse Bayesian formulation. An iterative reweighted ℓ1-minimisation algorithm based on the sparse Bayesian learning update is then derived to solve the fault diagnosis problem efficiently. With the proposed framework, a real-time fault monitoring scheme can be built using only measurements of phase angles at the buses.