What classical open problems are there in special relativity, including questions about non-inertial motion in flat-spacetime, but excluding questions about quantum theories.
Answers can include things which were open problems until recently.
New things in special relativity (and non-gravitational relativity) still seem to be discovered relatively recently:
For instance the book "A broader view of relativity" by Jong-Ping Hsu, Leonardo Hsu, contains formulas for transformations between some types of accelerated frames.
Abraham Ungar derived formulas for things like proper velocity composition, and for Lorentz transformation composition in terms of velocity and rotation parameters.
So it would be good to have a list of things that we still haven't worked out, or have only been worked out recently, to challenge the assumption that special relativity is all figured out.
(Due to linear frame-dragging it also seems that the usual saying that GR tends to SR in the limit of negligble gravity should be stated as GR tends to SR for negligble gravity and low acceleration, so any calculations concerning the half-way house of non-inertial relativity between SR and GR would only be valid for low acceleration. Perhaps this should be the definition of non-inertially extended special relativity: acceleration without linear frame-dragging.)
Edit: I +1ed the answer on self-force because it was an interesting general issue, but once we are talking about fields and particles then we are getting away from the question which is really about calculations in the theory of special relativity itself. Once we talk about fields and particles then I guess there are lots of problems and an accurate treatment requires quantum theories which this question is not about.