This thesis studies weighted projective planes and their connection to threefold singularities. In particular, we study the Veronese subring Sโ†’๐‘ฅ of the ring S associated with the weighted projective plane ๐• for choices of โ†’๐‘ฅ in the grading group ๐•ƒ. We show that there exists a projective, birational map Tโ†’๐‘ฅ โŸถ Spec Sโ†’๐‘ฅ under mild restrictions on โ†’๐‘ฅ. We then show that when โ†’๐‘ฅ = -โ†’ฯ‰, the dualising element, this map is a blow-up. In the toric setting, we show that in certain situations the singularities of S-โ†’ฯ‰ can be identified with the familiar cyclic quotient singularities and the map T-โ†’ฯ‰ โŸถ Spec S-โ†’ฯ‰ is a weighted blow-up. In particular, it is a crepant map. We also construct a tilting object on T-โ†’ฯ‰ in this setting. Away from the toric setting, we are able to construct tilting objects in some instances and we study some examples in depth to construct a full resolution and identify noncommutative resolutions of these singularities.