Honestly, not that I'm aware of in either SS or tube.  The only really noticeable instance perhaps would be overdriving the O2 when I first got it.  Then again, I do hear SS treble digititus or glare in many SS amps, I think there might be some debate about whether that is the same thing.  Again, that is a question wrt euphony versus transparency though it could said to be both.

The O2 is a special case and by special I might as well mean- retarded.

Also another common misconception would be to attribute the stereotypical tube sound to the active elements used. If I'm not mistaken the distortion pattern with low and even order harmonic content is inherent to single ended class A circuits. The Pass SS amps exhibit this kind of distortion and have often been associated with tube sound. I have heard the F4 and it sounds noticeably softer and lusher than a Bryston class-AB amp we had at hand for comparison.

Another common suspect for SS sound is the use of NFB and namely- the depth of the feedback loop. There are plenty of folks that regard feedback as an evil and strive for feedback-less amp and buffer stages.

If there is something that I would attribute to the thermionic valve as a control element- it's the superior linearity when dealing with low amplitude signals. Hence the plankton, microdynamics and holographic soundstaging that depends on small signal handling capability.

HOW much of 'the tube amp sound' will be present, due to added harmonics, depends on the circuit used as well as components such as transformers and capacitance of capacitors in relation with the load resistance (impedance) it is connected to.
Also the signal amplitude is of importance in non-feedback circuits.
When capacitors are too small in values they act as filters in the audible band and when they do certain properties may get in the way with proper reproduction depending on the choice of diëlectrics. This is touched by OJneg.

I'd really like to see some form of proof that all components (especially linear types like resistors or wires ) can produce any ringing below 100kHz (let's assume we can hear up to 100kHz).
Ringing is caused by feedback combined with propagation speed problems or inductances and capacitances creating a resonance point. Mostly waaaayy above the audible range.
Yes, in some tube amps where wirewound resistors are used you may get some overshoot but the choice of bifilar wound or carbon composite resistors may alleviate those problems, a component type choice rather than anything else.

Ultrabike has a point here:
For one, I'm not 100% convinced that a tube amplifier paired with an appropriate headphone load will yield more distortion than an SS amp. It may be that in some cases it's the opposite.

But as said before this depends on the circuits used and amplitudes present.

There is a very simple device called a differential amplifier (basically what's inside every opamp) which can show ANY difference between input and output (with the gain nulled out) and does so under actual load and with music signals. A very entertaining and eye opening device and if you ever get a chance to use one on a tube amp tell me what you are hearing. What is heard is what is changed (added or removed in amplitude, phase and time). Of course phase problems are the biggest contributors here and they aren't that audible in real life but become audible in this all revealing test.

The soft clipping issues of tubes are ONLY relevant when driven in clipping as is purposely done in guitar amps and is avoided in hifi amps. It is the reason why musicians love them... they call it tube sound and mean something totally different with that than hifi bufs do who usually do not come anywhere near clipping levels. If they do they need equipment with more headroom. Try using 2 of those guitar amps as a hifi amp or let a guitarist play via a hifi amp... that'l be fun.

As Firev1 already mentioned it is the harmonics distribution that counts. Tubes have higher 2nd HD and decreasing higher order distortions so third is smaller, 4th is even smaller and higher order drown away in the noise. SS (depending on the TYPE of components used) usually have harmonics higher up and thus the higher amplitude lower frequencies have 'unnatural' harmonics even in the treble area. As all signals have that the representation differs.
The harmonics distribution (decay in amplitude for the higher harmonics) of tubes is very similar to those harmonics that are present in natural instruments as most instruments generally do not exhibit higher order harmonics. So the harmonics from tube amps sound more natural or can enhance the sound to be more natural sounding. It also is cleaner up top in harmonics (becomes noise in that case)

The spread and amount of harmonics added depends on amplitude of the signal and the used components (as well as amount and type) and amount of feedback.
How audible they are depends on recording quality and resolution of the transducer (speaker/headphone)
A perfect amp doesn't add anything nor removes and is neutral. But amps with 'bad figures' may add pleasantness and more 'realism' for our puney brains. That last part is what N-W-AV-guy didn't get. He was only looking for 'near perfect& #039; reproduction and not to 'alter in a pleasant way'.
I see nothing wrong with the latter if it enhances the experience of perceiving music.

b.t.w. I am not a 'professional' amp designer but rather a (mostly) analog electronics designer and also design(ed) some amps (non commercial) over the years as well as serviced A/V for many years.

Exactly.  Before this site was even up, we ran a gamut of tests on tube amps and swapping different tubes.  You could blind test the various tube 'sounds' but when it came time to measure using the typical gamut of tests, they all measured essentially identical.  This is why the amp measurements went on the back burner, because they require a thorough reanalysis of how we objectively characterize an amplifiers performance.  We clearly need a more extensive suite of tests.  Hell, this was even true when we were able to blind test two SS amps w/ the same FR and ridiculously low distortion numbers, so it's not even just a tube thing.

So, has there been any progress (either here or elsewhere) on such tests to better characterize a DAC or amp's performance since then? Because even though my experience with and knowledge of audio equipment is quite small compared to a lot of folks on here, I have been able to hear differences in amps (ex: ODAC/O2 vs UHA-6S Mk.II with 209 opamp) even when they both measure extremely well and even similarly to each other. I think in the case of the O2/Leckerton, the O2 had a slight roll-off at the top and bottom ends of around 0.1 to 0.2dB if not even less than that. I believe the Leckerton was a bit less rolled-off at the top and bottom (source link below paragraph). I've seen some argue such small differences shouldn't be audibly perceptible, but I definitely heard a difference between the two DACs/amps. So, then, what's causing the difference, and can this be shown using known measurement techniques? Or was I really just hearing those very minute differences?

Source: http://scientistsaudio.blogspot.com/2012/06/leckerton-uha-6smkii-part-1.html

I've also heard the argument that there might be a reverb effect, which some tubes/amps can do but if the reverb can be isolated to an instrument and not color the entire track, how is that not better transparency into a recording rather than being colored as opposed to say many SS amps crushing entire tracks into a flat 2D plane?

I'd really like to know how >distortion can = >apparent transparency.  If that's not the cause, then what physical property about some tubes amps is?  I have a theory but I'd like to see where the Socratic method goes on this one.

Uh, well, according to what I've "learned" from the all-knowing HF members, your ears are definitely broken and/or are just hearing additional coloration...O2 is best, end of all discussions for the rest of eternity. lol