The former is a hypothesis that originated with observations of (apparent) periodicity in the extinction record, and has had its share of fame and controversy; the latter began as an innocent follow-on to the discovery of Pluto ('that puny little thing, a planet??') but in the late 20th century was kidnapped by a tribe of anti-science Californians. But can either be 'ruled out'? More precisely, in what regions of parameter space may such objects still lurk, awaiting discovery? Of the many techniques that could be used to discover any such 'hidden perturbers' (lovely phrase, donchathink?), which are the most powerful (in principle)? I mean this in the sense of 'straight-forward application with today's technology, at modest cost, could put much improved constraints on the existence of such objects in vast realms of parameter space'.