Honda is probably better known for its high-revving four-cylinder engines than anything else, and building a high-revving engine requires an obsessive focus on smoothness and balance. For Honda, that also usually applies to the rest of the car, a fact often noted by auto journalists back when Honda first hit big in the United States. But that strength can ironically prove an Achilles' heel after a few decades of wear and tear, when even minor vibrations snaking through the engine and chassis can turn into major aggravations.
The first step in diagnosing vibration problems at idle is to watch engine rpm. After 20 years of operation, there's a pretty decent chance that one or more of your engine mounts is worn out. This is a pretty common failure for these cars. If your engine rpm is holding steady at 900 to 1,000 rpm at idle, and you still have vibration, then odds are best you have a bad motor mount. This can make minor idle issues more obvious, especially if you have a dirty throttle body or idle-air-control circuit. Again, these are common occurrences on cars this old, and can easily cause an idle fluctuation and slightly rough idle. Combine that with somewhat worn motor mounts, and you have a disconcerting grumble at idle.
Cylinder misfires are almost always at the root of engine-related vibrations, and any number of things can cause them. Even with a pre-OBD2 car like this, you should get a check-engine light and trouble code if there's anything wrong enough with the engine to cause a cylinder misfire. A bad ignition coil, spark plugs in need of replacement, low fuel pressure, a clogged filter or clogged or malfunctioning fuel injectors will typically cause a power loss and misfire at higher rpm and under acceleration. Vacuum leaks will cause the engine to run rough, but this will generally smooth out at higher rpm. Otherwise, any one of the engine's sensors could malfunction and cause a misfire, and you'll almost certainly get a check-engine light with any of them.
If the source of your vibration under acceleration isn't the engine, it may be the transmission. Worn-out transmission fluid, worn-out regulator springs in the transmission or low fluid pressures can cause the clutches to shift, leading to vibration under acceleration -- particularly when shifting. Odds are better though that, if it's not the engine or motor mounts, your CV joints are to blame for vibration under acceleration. This failure is notable because it will also cause vibration under braking, and you'll generally hear a grumble or click with the vibration. Bad outer CV joints will also produce a more noticeable clicking, grinding noise and vibration while turning. Vibration under braking may also be warped or worn-out rotors if the vibration only happens under braking -- a fairly common failure on these cars.
Low-speed vibrations that don't come from any of the above-mentioned causes usually go back to a failure in the suspension or steering system. Oddly, worn-out steering end-links and tie rods will probably vibrate worse when pointed straight then while turning, as turning tensions the links and eliminates some of the slop. A bubble in a tire will cause a regular, pronounced vibration that's always present and gets faster with speed. Many times, though, you'll get no vibration at all until you get to a certain speed; this may go back to an imbalanced wheel or a damaged tire. Bad wheel bearings will also start to vibrate, grumble and squeal over a certain speed. Worn or broken suspension components, ball joints or struts will typically make some noise over bumps and road irregularities. A worn-out or blown-out shock absorber will exacerbate the vibration from a slightly out-of-balance wheel, but you should notice a ride quality change or excess dive under braking if this is the case.
