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We study the integral model of the Drinfeld modular curve $X_1\left(n\right)$ for a prime $n\in {𝔽}_q\left[T\right]$. A function field analogue of the theory of Igusa curves is introduced to describe its reduction mod $n$. A result describing the universal deformation ring of a pair consisting of a supersingular Drinfeld module and a point of order $n$ in terms of the Hasse invariant of that Drinfeld module is proved. We then apply Jung-Hirzebruch resolution for arithmetic surfaces to produce a regular model of $X_1\left(n\right)$ which, after contractions in...