According to theories of quantum gravity, spacetime may be non-fundamental. The implications of this observation are now widely debated in the philosophy of quantum gravity. In this paper we argue that what is often discussed under the umbrella term of `spacetime emergence' in the philosophy of quantum gravity literature in fact consists of a plethora of distinct and even highly different problems. We therefore advocate to cast such debates more specifically in terms of emergent spatiotemporal aspects as is already done (...) in the physics literature. We first show how ambiguous the notion of spacetime is and offer five understandings of what the problem of spacetime emergence may still amount to. We then argue, however, that there are many philosophical problems relating to spacetime emergence and that none of the five understandings picks out a problem that is exceptional among these. Next, we observe that different spatiotemporal aspects are emergent in different quantum gravity approaches whereby speaking of quantum gravity collectively is problematic. Finally, we illustrate how inquiries about spacetime emergence are actually aided by conducting the investigation at the level of specific spatiotemporal aspects. (shrink)