State of the art in arithmetic moduli of elliptic curves? In trying to get into the topic of moduli spaces of elliptic curves, the following question arises:

What is the state of the art in the topic right now?

Deligne and Rapoport describes how the moduli problems with level-$N$ structure each have a model over $\mathbb Z[1/N]$. This is according to Igusa. Next, in the preface to Katz and Mazur's book, some work of Drinfeld is mentioned and the book's aim is to apparently re-write the Deligne-Rapoport construction based on Drinfeld's work. 
Here I am a bit confused. What does this given in addition? There is already a smooth model over $\mathbb Z[1/N]$; so what does Katz-Mazur give in addition?
Next, are there any advances in this topic after Katz-Mazur?
 A: Your sketch of the history is rather misleading: you make it sound like everything in Katz--Mazur is a rewrite of Deligne--Rapoport, and everything in Deligne--Rapoport is a rewrite of Igusa! As far as I (not an expert) understand, the main contribution of Katz--Mazur was to define models over $\mathbf{Z}_p$ for $p$ dividing the level, which are not smooth of course, but still have some good moduli interpretation (in terms of Drinfeld level structures). Deligne--Rapoport don't do this: they do define a $\mathbf{Z}_p$-model for $X_0(p)$ but it doesn't represent a functor in any obvious way, and they don't do higher $p$-power levels. 
As for current work in the area: there is a lot of literature on finding semistable models of modular curves (e.g. Coleman and McMurdy have a series of papers on this); and there is a lot of very deep work extending the results from modular curves to more general Shimura varieties, e.g. Kai-Wen Lan's Harvard thesis (published as a monograph "Arithmetic compactifications of PEL Shimura varities").
