Fully linear time regular expression matching
Groz et al. explicitly state that the best known algorithm for general regular expressions (as of 2012) is $O(nm(\log\log n)/(\log n)^{3/2}+n+m)$, due to Bille and Thorup 2009, doi:10.1007/978-3-642-02927-1_16 (preprint).

For a fixed size alphabet, Sebastian Maneth pointed out to me that $O(n+m)$ is possible for deterministic regular expressions by constructing the Glushkov DFA: each position in the regular expression is one state, and the transitions are determined by the bounded set of symbols that may appear before moving to a position.
However, without fixing the alphabet size, it still holds that "finding a time $O(m+n)$ algorithm remains an open problem" even in the deterministic case.