Positroid varieties are subvarieties in the Grassmannian defined by cyclic rank conditions and which are related to Schubert varieties. We will provide a criterion for whether positroid varieties are smooth at certain distinguished points, and we will show that this information is sufficient to determine smoothness for the entire positroid variety. This will involve looking at combinatorial diagrams called "affine pipe dreams." We can also form a partial order on positroid varieties given by deletion and contraction, such that there is closure for smooth positroid varieties, and we will characterize the minimal singular elements in this order. Finally, we will discuss a couple of connections between the techniques of this work and planar N=4 SYM: the BCFW bridge decomposition and inverse soft factors.