I'm not somebody who has ever had a deep involvement with the subject, but I have had a few skirmishes.
One was when I was running the approval testing of a modified Tucano T1 for flight test use. We'd fitted a big instrumentation boom at about half span on the port wing, which was there to provide slipstream-free AoA and Beta. Initial tests (on which I hadn't flown) showed that the data was completely useless - all over the shop. In an effort to try and understand what was going on, I ran the AoA data through a digital FFT (Fast Fourier Transform) analyser, and clearly showed a continuous resonance which was queering the results - from memory I think that it was somewhere between 10 and 1 Hz. I then went rear seat for a sortie (looking at other stuff as well) and noticed visually that there was some tendency of the boom (which was easily visible from the rear seat). Putting two and two together, I ordered a "bonk test" to be carried out on the boom on the ground.
At which point one should explain what a bonk test is. This is a method by which the resonant frequencies of discrete parts of an aeroplane can be determined. The part is fitted with a very lightweight high rate accelerometer feeding an equally high rate data-logger. You then hit the item with a calibrated soft faced hammer (actually it's usually a device firing a little rubber encapsulated bullet nowadays). So, when you hit the aircraft part, it resonates (one hopes briefly!), this is recorded by the data logger, and you can then run that data through an FFT analyser - this tells you what the resonant frequencies are of the part. You obviously don't want this to, say, co-incide with any of the normal cruising RPM values of the engine.
In this case, to my unsurprise, the main resonant frequency of the boom/head combination was co-incident with the frequency that I'd dropped out from my FFT analysis of the AoA data.
So, I then went away and did a simple SHM (Simple Harmonic Motion) analysis of the boom as a mass-spring system, from this I could drop out some assumptions about the spring constant (mass was easy, I just took it off the design drawings, but I wasn't in the mood to spend days doing a probably inaccurate estimate of the spring characteristics of a reasonably complex internal shape). A bit of playing around with numbers showed that if we shortened the boom by 6" it should be fine. So, we did that, and went and tested it - the AoA trace was steady as a rock, and the design got signed off. To the best of my knowledge, it's still flying and providing ETPS students with much material for writing up longstab and lat-dir reports.
The timescale of this was fairly short - a few weeks within another (bigger) task. I don't think it's unrepresentative of most ground resonant testing that gets done, which is usually alongside other tasks, and supported by analysis and flight test results - basically looking for and solving problems that you know are there already (or at the very least, anticipate).
Watch this space, and I'll follow up when I've time with some tales of vibrating beaver tails and spontaneously combusting propellers, both of which were related to resonance and required ground testing and aircraft design changes.
G