I will discuss forthcoming work which constructs phantom categories in the derived category of the blow-up of P^2 in 11 general points and the Hirzebruch surface F_2 blown up in 9 general points. I will compare these results to existing ones in the literature and then present a conjectural description of which rational surfaces admit phantom categories.

This is based on joint work with Kimoi Kemboi, Daniel Krashen, Tianle Liu, Yeqin Liu, Eoin Mackall, Svetlana Makarova, Alexander Perry, and Sridhar Venkatesh.