Which elliptic curves over totally real fields are modular these days? As the title says.  In particular, every elliptic curve over $\mathbb{Q}$ is modular; but what is the current state of the art for general totally real number fields?  I assume the answer is extractable from some papers of, say, Kisin, but I am not an expert in this material and hesitate to try that myself.
 A: Elliptic curves over real quadratic fields were proven to be modular very recently by Freitas, Le Hung and Siksek:


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*Nuno Freitas, Bao Le Hung, Samir Siksek, Elliptic Curves over Real Quadratic Fields are Modular (2015)


Some progress was alredy present in the thesis of Richard Taylor's student Le Hung.


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*Bao Le Hung, Modularity of some elliptic curves over totally real fields (2013)


Potentially modular is known unconditionally over arbitrary totally real fields, that is:

Theorem. Let $E/F$ be an elliptic curve over a totally real field. Then there is some totally real extension $F'/F$ such that $E/F'$ is modular.

As mentioned in the comments, for practical purposes this is almost as good as modularity.
The attribution of this theorem is complicated. There's some information about this in section 7 of our own Kevin Buzzard's very interesting survey on modularity:


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*Kevin Buzzard, Potential modularity—a survey (2011)


You can also consult:


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*Jean-Pierre Wintenberger, Appendix: Potential modularity of elliptic curves over totally real fields (2010)


For arbitrary totally real fields, modularity is still not known.
