It heavily depends on soil conditions, slope of the trail, relief of the terrain, frequency of precipitation, and freeze/thaw cycles.
I have not ridden clay-based soil anywhere that did not respond poorly to precipitation of any kind. Small particle sizes absorb a lot of moisture and don't infiltrate rapidly. Some types of clay even expand/contract SIGNIFICANTLY as soil moisture changes.
If the trail is mostly rock with some clay between the rock, things seem to do better where there is more rock content. But there will still be places where clay content is very high (like the bottoms of the hills) and those places will still respond poorly.
Sandy soils respond much better. They have relatively larger particles and water infiltrates rapidly. The larger particles mean for larger pore spaces which keeps the water moving. Sandy soils do not retain moisture well so they can be ridden in wetter situations (and in fact, many are better when they're somewhat wet).
Heavily organic soils respond poorly. Organic material acts like a sponge. In some places with heavy organic content, the soil remains wet all the time.
Hilly terrain can mitigate some factors. If a trail is on a poor quality soil, but it is located on terrain that drains water rapidly via sheeting, the trail can handle rain better.
Trails in floodplains are especially difficult. You've got typically poor relief, you're very close to the water table, you frequently have high organic content, and there are frequently small size particles in the soil (silts and clays). Don't play with water on these trails EVER. It's always bad.
You also need to keep in mind freeze/thaw cycles. The same process that creates potholes in the pavement also heaves soil, which can partially decompress the packed soil on trail tread. When it thaws out again, it has increased capacity to retain moisture and the mud is especially bad. But what can be worse is when only the surface of the soil is thawed, but the lower layers are still frozen. That frozen layer acts as a barrier to moisture, preventing the soil from draining and drying out. Then you have to keep in mind that the plants are not yet absorbing moisture from the soil so the soil has to drain even more moisture.
There are a lot of variables if you want to approach the situation from an analytical perspective.
Sometimes, it's better to keep things simple. If there's mud on your bike, it's too wet to ride. But, it can also be too wet to ride, but the mud does not stick to the bike. In those situations, you'll still be leaving quite a rut. I've heard guidelines that if you leave a rut at all, don't ride or if your rut is deeper than 1/2", don't ride. I think those are all good to keep in mind.
