When I was getting my pilots license I stumbled into a really cool video NASA did showing exactly what happens when you reach a fluttering condition like what you described. This is a link to the NASA recording if anyone is interested.
In practical terms, flutter due to approaching transonic conditions is probably pretty rare. That kind of flutter is (mostly or most often) due to the interaction of normal shocks with the control surfaces, or coupling with an elastic mode of the wing or tail surface.
The flutter in the first video is occurring at very much subsonic speeds, and looks to be either the result of flying a purposely underdesigned tail surface, or flying a properly built one beyond its rated flight envelope. The second video contains a wide variety of flutter instances, some of them aeroelastic, some of them transonic, and so on.
One way to get into a transonic flutter, however, is to be a hotshot business jet pilot who flies higher and higher and faster and faster. The higher you go, the lower the density, so your minimum speed increases. Also, the temperature goes down, so the speed of sound decreases. Where these two meet is called "coffin corner," and you don't always have to fly yourself into it by increasing your speed and altitude; you can fly close to coffin corner, and then fly into colder air or less dense air. No matter how you get there, you're stuck. Slow down, and the wings stall, the nose drops, you pick up speed, and hit transonic flutter. Speed up to stay in the air, by dropping the nose, and you hit transonic flutter directly.
Do you have any thoughts on composite-based aircraft like a Cirrus and flutter? A few times I've had a Cirrus SR22 into a pretty steep descent with poor controller sequencing for an approach into busy terminal space and had to push it down, but the plane felt solid even at 180-190kts TAS. I backed it off only because I get nervous with any unexpected turbulence which is not uncommon in Florida.
The Piper Saratoga I flew for a bit didn't seem to like the speed as much, that or the toga was a bit more vocal than the Cirrus in what it was feeling with regards to airspeed.
Only in the most general terms, and from first principles: Composite structures will have a higher stiffness per unit mass, which will cause the fundamental frequencies to be higher both for the pure structural modes and the control surface interaction modes. It's therefore likely that you'd only begin to encounter these aeroelastic modes at higher speeds. In other words, if you take the driving frequency as something like the inverse of the time it takes for air to pass over the wing, that may match the metal wing more closely than the composite wing. Modeling that stuff in a wind tunnel is very tricky, and you used to end up with these not-at-all-realistic-looking models that, nonetheless, captured some aspect of the full-scale aircraft being modeled.
Don't die. Stay in the envelope. Flutter is only the quickest way to ruin your airframe and day, not the only one.
https://www.youtube.com/watch?v=pEOmCkZyXzk
https://www.youtube.com/watch?v=OhwLojNerMU (Older with music, kinda funny old-school science video from NASA)
Edit: formatting.