An example of Serre shows that in the strong form of his modularity conjecture, one can't always ask for minimal nebentypus. Serre and Carayol independently explained that this obstruction is due to nontrivial isotropy groups on certain modular orbifolds, hence only occurs for the primes 2 and 3 and certain Galois representations called badly dihedral.
Curiously, when studying the deformation theory of a mod p modular Galois representation for an odd prime p, the same badly dihedral representations for p = 3 arise: it is exactly for these that the minimal deformation ring does not appear to be a flat local complete intersections over the ring of Witt vectors.
We explain this link via a derived version of a minimal R = T theorem. As a corollary, we can characterize when these badly dihedral representations admit lifts with minimal weight, level, and nebentypus. This is joint work in progress with Preston Wake.