Experimental Verification of No Special Frames of Reference Certainly, there have been numerous tests of both Special Relativity and General Relativity.
Given all the phenomena and behavior stipulated by Relativity, one could perhaps divide these phenomena into two categories:
Things the don't depend on "No Special Frame of Reference":


*

*Time Dilation

*Length Contraction

*Lorentz Covariance

*Pretty much all of General Relativity


And things the do depend on "No Special Frame of Reference"


*

*All frames see other frames' clocks moving slower

*All frames see other frames as length contracted

*Relativity of simultaneity


It seems that a lot of the particular mind bending aspects of Relativity arise from trying to maintain the "No Special Frame of Reference" rule.  As far as I can tell, all of the experimental verifications of Relativity involve items contained in the first list.  Is this the case?  Have their been experimental results that verify any of the items in the 2nd list, that is to say that verify the "No Special Frames of Reference" rule.
Requested Clarification
Relativity says that under certain circumstances clocks will dilate.  This has been verified.  Relativity also says that if two frames of reference pass each other both will see the other's clock as moving slow as opposed to both frames seeing one clock being fast and the other clock being slow.
I believe this assertion follows from the notion that there are no special frames of reference.  Has this particular aspect of relativity been experimentally verified?
 A: For Special Relativity (SR) i think the Michelson-Morley experiment is compatible and provides a verification of SR principle (some other formulations are also compatible with the experiment).
Quantum Field Theory and especially the Dirac prediction and verification of positron is also a verification of SR (and many other expreriments in this context)
For General Relativity (GR), this is not exactly the case. Of course GR had 3 crucial tests, a) Mercury Perihelion, b) Red shift and c) Light-deflection by the Sun but the question is about the general covariance principle which to my knowledge is not verified fully (and some Quantum Gravity approaches do not incorporate it at least as Einstein proposed). Other principles of GR like the equivalence principle is also verified experimentaly.
An older survey by L. Schiff on Tests of Special and General Relativity
A: Your question exposes the importance of defining notions in physics unambiguously and universally in terms of "How to measure?".
As Einstein put it explicitly (however, referring specificly only to the notion of "simultaneity", and unfortunately only as late as 1917):
"We thus require a definition of simultaneity such that this definition supplies us with the method by means of which, in the present case, he can decide by experiment whether or not both the lightning strokes occurred simultaneously. As long as this requirement is not satisfied, I allow myself to be deceived as a physicist (and of course the same applies if I am not a physicist), when I imagine that I am able to attach a meaning to the statement of simultaneity. (I would ask the reader not to proceed farther until he is fully convinced on this point.)"  [Translation of the German original retrieved from http://en.wikisource.org/wiki/Relativity:_The_Special_and_General_Theory/Part_I#Section_8_-_On_the_Idea_of_Time_in_Physics]
Likewise, obviously, the notion of "(joint membership in one) inertial frame", a.k.a. "(mutual) rest" is meaningful only as far as a method has been declared and understood for deciding case by given case whether and to whom this characterization applied. 
Only once the corresponding method has been provided and agreed on we can think concretely of
1. applying it to available observational data, and thus
2. experimentally testing any (pre-existing) models or expectations about "who was at rest to whom, in which trial".
Any underlying methods ("How to measure whether ...") are of course not themselves subject to experimental testing.
This requirement of "definition through a method by which to decide" must certainly be fulfilled by all notions which are intended to "differentiate experimental settings":
for instance, surely we wouldn't require by definition that all distinct participants should be "at rest to each other";
nor that all distinct members of an inertial frame should have "equal pairwise distances from each other";
nor that all distinct oscillation periods of some oscillator under consideration should all have "equal duration"; etc.
In these cases we need to declare and understand "How to measure whether ...".
But it should also be obvious that the requirement of being "defined through a method" cannot be upheld for all notions whatsoever. There must be certain sufficient notions recognized (presumed, or, in turn, granted) by which to express any further "methods" in the first place; i.e. notions which are considered self-evident (a.k.a. "axiomatic") and which therefore guarantee that the methods to be expressed are indeed unambiguously and universally comprehensible and applicable.
On this, Einstein also left a clue, namely:    
"All our well-substantiated space-time propositions amount to the determination of space-time coincidences [such as] encounters between two or more recognizable material points."    [Translation of the German original retrieved from http://en.wikisource.org/wiki/The_Foundation_of_the_Generalised_Theory_of_Relativity#ch.3.p.776]. 
Before drawing any further conclusions, the task at hand for physicists (and with relevance to non-physicists) is therefore to use these axiomatic notions in order to express (and agree upon) some method for experimentally deciding "(joint membership in one) inertial frame", a.k.a. "(mutual) rest", in the first place. (A sketch of my own attempt is documented in my answer to the question "What determines which frames are inertial frames?" (PSE/a/70646).)

Remark added in response to a comment:

Certainly, there have been numerous tests of both Special Relativity and General Relativity.

Hardly. The very idea of wanting to have "experimental tests" of Special Relativity and/or General Relativity is utterly mistaken. The Theory of Relativity, as proposed by Einstein and elucidated especially by the above quotes, is a system of axioms, definitions (to be expressed by means of the axiomatic terms), and resulting theorems, concerning how to measure geometric and kinematic relations (between identifiable participants, a.k.a. "material points");
in particular dealing with how to measure 


*

*whether and which given participants had been "at rest" to each other (a.k.a. joint members of one "inertial frame"), 

*geometric relations between participants who had been at rest to each other (namely: values distance ratios), and

*kinematic relations between participants who had been at rest to each other, but (especially, foremost) who had belonged to distinct inertial frames (namely: "speed ratios $\beta$").
But RT does not involve any experimentally testable hypotheses or models that any particular "material points" under consideration, in any particular trial under consideration, had had any such specific geometric or kinematic relaions. RT merely deals with the methods of how to measure whether; but those are not themselves experimentally testable, instead they are necessary pre-condition for expressing any experimentally testable hypotheses or models dealing with geometric or kinematic relaions in the first place (such as any experimentally testable models, and especially the so-called "standard models", of cosmology, of astronomy or astrophysics, of material science and chemistry, of particle physics -- you name it). 
Note that the whole of RT (axioms, definitions of measuring methods, etc.) has remained and continues to remain intact and useful and uncontroversial even if any particular experimentally testable hypothesis or model about geometric or kinematic relaions of given "material points" had been found false (experimentally falsified) by using those very measuring methods of RT. 
