TITLE: Which elliptic curves over totally real fields are modular these days?
QUESTION [22 upvotes]: 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.

REPLY [9 votes]: Elliptic curves over real quadratic fields were proven to be modular very recently by Freitas, Le Hung and Siksek:

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.

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:

Kevin Buzzard, Potential modularity—a survey (2011)

You can also consult:

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

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