Open problems in special relativity, and non-inertial motion in flat-spacetime, or things that have been discovered recently 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.
 A: I can think of the problem of a self-force on an accelerating charged particle as one that still needs a full solution.
Another important question in classical relativity is that of the motion of extended bodies in curved spacetime. Turns out that because the motion of an extended body in a curved space appears differently to different observers, such an object can actually translate ("swim") through space by periodic distortion of its shape [reference].
There are other open classical problems in relativity but again, as Piotr, said its hard to think of any in flat space.
A: I can tell you how to generate open problems in special relativity: open up a geometry book and look at some geometry theorem. Add the word "Minkowski" before everything. Is the theorem still true? One problem that I got to thinking about recently is how to axiomatize Minkowski geometry in the same way that Hilbert axiomatized Euclid's geometry. It is a theorem that Euclidean geometry (as axiomatized by Hilbert, essentially completing Euclid's postulates) is a complete theory: there is an algorithm for deciding which statements are theorems and which are not. Is the same true of Minkowski geometry?
A: Would you consider something like finding a classical solution to the Yang-mills equation of motion a problem in special relativity?  Because I think that's still open for non-$U(1)$ symmetry groups.  
A: Special relativity is rather simple. Abraham Ungar's stuff that you are quoting seems quite trivial at first (and second) glance. There is no difficulty combining arbitrary rotations and Lorentz boosts. You only need to express those as 4x4 matrices. I derived such matrix one time simply to answer a forum post.
The linear frame dragging results from motion of mass, and as such is, too, a gravitational effect. The general relativity is rather complicated and there are various unsolved problems.
A: "Accelerating relativistic reference frames in Minkowski space-time" arXiv:1109.1796, 9 Sep 2011
Slava G. Turyshev, Olivier L. Minazzoli, Viktor T. Toth
A: The Einstein relativity is the viewpoint as seen from moving bodys. Since all objects are in motion we think that there is no need to study any other viewpoint. Is that all? 
One subject that has been negleted is the one-way/two-way speed of light.
In Einstein SR c is the mean light speed in a closed path and it is constant to any moving observer, as it is. 
My comments on this: One-way_speed_of_light#Preferred_reference_frame 

A preferred reference frame is a reference frame in which the laws of
  physics take on a special form.   

Why?  

The ability to make measurements which
  show the one-way speed of light to be different from its two-way speed
  would, in principle, enable a preferred reference frame to be
  determined. This would be the reference frame in which the two-way
  speed of light was equal to the one-way speed.

This is about the CMB referential! It is unique and is common to all moving bodies and where the speed of light is isotropic. Since we are not able to measure the one-way light speed, we are persuaded that there are no effects.
How sure we are if we dont have the relevant studies?
IMO...
Afaik the computation of energy is frame dependent. Can I think that the neutrino-(originally a missing energy problem) do not exists in the absolute referential?
IMO, the flyby anomaly, chirality, matter/antimatter, and a lot of other unsolved issues can have an explanation if we dare to make the studies.  

In Einstein's special theory of relativity, all inertial frames of
  reference are equivalent and there is no preferred frame. There are
  theories, such as Lorentz ether theory that are experimentally and
  mathematically equivalent to special relativity but have a preferred
  reference frame. In order for these theories to be compatible with
  experimental results the preferred frame must be undetectable. In
  other words it is a preferred frame in principle only, in practice all
  inertial frames must be equivalent, as in special relativity.

We are avoiding to aknowledge that the CMB reference is really special. IMO we are poisoned with prejudice.  
In this paper Cosmological Principle and Relativity - Part I (2002, not peer reviewed, by my friend Alfredo that was fighting against a cancer in those days) is presented a study 'from above'. This work enable us to start to understand what kind of nuances can appear, namely in Lorentz length contraction formula, if we study light and motion from the CMB viewpoint. Einstein SR equations are also derived as a by-product.
In conclusion: Einstein viewpoint is correct but it is not all the story.  
A: Length_contraction#Experimental_verifications
Length Lorentz contraction - observer frame dependent ? or ...
Length Lorentz contraction - absolute, i.e. real  ?  

Since the occurrence of length contraction depends on the inertial
  frame chosen, it can only be measured by an observer not at rest in
  the same inertial frame, i.e., it exists only in a non-co-moving
  frame. ...
  Another confirmation is the increased ionization ability of
  electrically charged particles in motion. According to
  pre-relativistic physics the ability should decrease at high speed,
  however, the Lorentz contraction of the Coulomb field leads to an
  increase of the electrical field strength normal to the line of
  motion, which leads to the actually observed increase of the
  ionization ability

How explain this "observed increase of the ionization ability" as a frame dependent measure?  IMO, this points to an absolute effect. 

see the explanation of the Real Lorentz–FitzGerald contraction (by Hans de Vries) 
from his online book:  
