I have a 2008 Ford Focus with 122k miles in good condition, nothing ever replaced except an engine mount, shocks, and coolant thermostat and all regular maintenance.
In the last four months I have had a few instances of the car shaking during my commute. Only happens when I am either stopped at an exit ramp or stopped during heavy traffic on the highway. It can happen anywhere from 10-30 minutes into my commute. Feels like either the road is shaking from an earthquake or godzilla is standing behind it and pushing it back and forth. It’s very strong but very smooth and slow. Like being on a boat. There is no vibration in the steering wheel or any other parts and no noise associated with it. It goes away once I start driving again.
I had a cracked engine mount once that produced a lot of vibration and rattling, and this shaking doesn’t feel or sound anything remotely like that one, so I am doubtful that it has the do with the mounts (just had that one replaced a couple months ago anyway).
This is also extremely intermittent. Maybe once a month at most, although the time span in between occurrences seems to be shorter so that’s why I am investigating. And when it happens, it’s just once during my entire drive and then not again for another few weeks or so. So it would be hard to take it into a mechanic to reproduce.
What could this be? I’ve been thinking about getting a new car and this has me worried that I’m entering into the timeframe with this one where major things start breaking.
So this only occurs when the engine is idling and you are stopped? But moving, it is fine?
Did you try shifting into neutral when this occurs? blipping the throttle?
I have trouble visualizing the car shaking so much that it feels like an earthquake with the engine idling…
There was no mention of a lit-up Check Engine Light.
Is that light lit-up, or–even more seriously–is it flashing when these shaking incidents take place?
Sounds like it’s running on three cylinders. When were the spark plugs last changed?
Well, it’s an inline 4 (all gen2 Focus have some flavor of I4). For I4s, a small amount of vibration is normal; it’s usually only noticeable at low RPMs. The I4 layout is inherently unbalanced; the engineers compensate for that with counterweights, but it’s not perfect. If it’s that bad, I’d suspect these counterweights may be loose. Worth a look.
Happens only when stopped. Have not tried shifting into neutral. It doesn’t happen often but I can try next time it pops up. The earthquake is an exaggeration. 
But it is surprisingly strong shaking, hard to believe it’s coming from my car considering there’s no noise or vibration associated with it.
Check engine light is not on. I just had it in the shop too for a thermostat change, they didn’t see anything else wrong and no other codes came up besides the thermostat.
Spark plugs have not been changed since I bought it at 36k miles. Could that be it?
I think you may have one or more spark plugs that have worn far enough for the gap to be too wide for the spark to jump. The spark is slightly weaker at an idle than when the engine is running at cruising speeds.
I’d just take it in and get a set of new plugs, they have gone enough miles and are due.
Ask your mechanic to also keep a sharp eye for any other things that could be causing your problem.
Yosemite
"Spark plugs have not been changed since I bought it at 36k miles. Could that be it?"
I would be very surprised if the specified interval for changing those plugs wasn’t–at most–100k miles. Do yourself a favor and take a look at the maintenance schedule that should be contained in the Owner’s Manual.
Even if changing over-aged spark plugs doesn’t immediately resolve the current issue, it is important to replace them if you want to continue driving the car without breakdowns and with maximum fuel efficiency.