It's very hard to imagine that there is any sensible model consistent with OPERA's results. (Aside from models of unaccounted-for systematic uncertainties in the experiment.) We know that we live in a world described to very high precision by Lorentz-invariant quantum field theory, so the most sensible way to look for Lorentz violation is to start with such a theory and add small Lorentz-breaking terms to the Lagrangian. Because we assume that they're small, the usual dimensional analysis that tells you if operators are relevant, marginal, or irrelevant in effective field theory still applies. So, for instance, instead of a particle having a kinetic term $\eta^{\mu\nu} \partial_\mu \phi \partial_\nu \phi$, we can add a piece $\epsilon \partial_t \phi \partial_t \phi$ that picks out a preferred time direction. This continues to be dimension 4, so it's a marginal operator -- you should expect such operators to be present with order-1 coefficients, and they should run logarithmically just as in usual field theory, so if you try to set all of them to zero except the ones for neutrinos, say, the others will be generated nonetheless. This is already a bad sign for the viability of such theories.
The Lorentz-violating extension of the SM, defined along these lines, has been studied by Coleman and Glashow and by Colladay and Kostelecky. (As Lubos mentions, Kostelecky has done a lot of careful and serious work on understanding exactly how well Lorentz invariance has been tested.) It turns out that there are not just marginal operators, but also relevant ones, which makes the whole picture look even worse. However, the relevant operators all violate CPT, so if you decide to consider only theories that violate Lorentz invariance while preserving CPT, you're on slightly safer ground, although marginal operators are not by any means safe. The bounds are extremely strong: a 2008 review by Kostelecky and Russell has convenient tables of various operators. From there you can find references to other papers like this one of Altschul that derives a bound of about $10^{-11}$ on the analogue of "$\epsilon$" for muons. If you want a theory where muon neutrinos see a violation of the speed of light, you'll inevitably generate such a thing for muons -- even if you cleverly try to arrange it to depend on electroweak symmetry breaking to evade the fact that they live in the same doublet, loops will inevitably generate the term for muons, at a level larger than $10^{-11}$. Electrons are even more strongly constrained, at the $10^{-15}$ level, and because neutrinos oscillate into different flavors, you can't avoid confronting that bound as well.
That's not to mention the supernova 1987A constraints on neutrinos, which tell you that if you want all this to work you need to go to even more exotic theories and look at energy dependence, as you noted in the question. You can try to do that with some higher-dimension operators, but again, effective field theory tells you that you can never do that sort of thing in isolation. Break the symmetry somewhere, and all allowed terms are generated. And they're all very well bounded by data. (The size of the effect tells you that these higher-dimension operators would be suppressed by relatively low scales, which is also not encouraging.)
I won't claim that I've given you a completely airtight argument here, but anyone who really wants to claim to have an explanation of how a particle can go faster than $c$ has to confront these effective field theory issues. Anyone who claims to be able to just sidestep them completely is either trying to fool you, or fooling themselves.
Edited to add: Because I think Moshe is right that this point isn't widely enough appreciated, while I'm on my soapbox I might as well point out for readers not well-versed in effective field theory that precisely the same argument should be deployed against the idea that physics is fundamentally discrete, or that we might be living inside a condensed matter system that flows near a Lorentz-invariant fixed point, or all sorts of other new kinds of science that people might try to sell you.
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