Neural information processing in dynamic, natural environments requires the brain to flexibly allocate limited computational resources to varying task demands. One example is selective attention in the visual system, which allows to preferentially route signals from behaviourally relevant stimuli to downstream visual areas, while suppressing irrelevant visual information. It is an open question which neural mechanisms realize selective routing, and how this process is controlled. Here we propose that spontaneous synchronization in recurrent networks is the key mechanism for selective processing. We study a hierarchical network consisting of recurrently coupled populations of spiking neurons which send activation to a common receiver/output population. The network is driven by two external signals of which one has to be routed to the output, while the other signal has to be ignored. Communication takes place via propagation of spike avalanches between sending and receiver populations. Routing is established by releasing inhibition from control populations which enhance avalanche generation and increase sizes of synchronous events carrying the information from the attended stimulus. Our framework provides a unifying account for selective information transfer through the visual hierarchy while reproducing a series of key experimental observations, such as typical rate modulations induced by attention in different visual areas, and the emergence of gamma oscillations and inter-areal phase synchronization. In contrast to previously proposed routing schemes based on oscillatory dynamics such as Communication-through-Coherence, selective routing can be established quickly since it does not need an intricate control scheme organizing the phase relationship between different oscillatory units. In addition, it proposes a simple and biophysically plausible control mechanism in form of a population with a critical dynamics, which becomes engaged by attention and then boosts generation of synchronous events for enhancing signal transfer.