Suppose that $X$ is a non-singular variety and $Z \subset X$ is a closed subscheme. When is the
blow-up $\operatorname{Bl}_{Z}(X)$ non-singular?
The blow-up of a non-singular variety along a non-singular subvariety is well-known to be non-singular, so the real question is “what happens when $Z$ is singular?" The blow-up can be singular as the case when $X = \mathbb{A}^{2}$and $Z$ is defined by the ideal $(x^2, y)$ shows. On the other hand, the example where $Z$ is defined by the ideal $(x,y)^2$ shows that the blow-up can be non-singular.
Edit: Based on the comments of Karl Schwede and VA, I think that it would also be interesting to find non-trivial examples of appropriate $Z$'s. I am splitting this off as a separate question. In the comments there, the users quim and Karl Schwede say a bit about what can be said about this question using Zariski factorization.
Best Answer
Craig Huneke told me about this paper: "On the smoothness of blow-ups" (MR1446135, Zbl 0878.13004, by O'Carroll and Valla). The title alone seems to suggest it might be useful for you.