How would you tell, either from dismounting or from horseback if it's safe to jump an obstacle?
Can you always assume a cross country obstacle will be safe to jump as well?
Firstly, the best way to test if an obstacle (commonly called a jump) is safe to jump is to walk around it and give it a once over. This can be mounted or dismounted leading the horse.
What you want to look for is holes or unevenness in the ground - usually caused from jumping in the wet and the ground drying with deep horse prints in. You also want to look at the quality of the jump, is the wood (log, or telegraph pole etc) splintered badly? If so a miscalculation in the horses jump could lead to a cut. Rotten jumps can also be an issue, but usually they should be fine.
Also worth bearing in mind is if the jump has a clear line between height and floor, but this I referring to the fact that a single bar floating in the air is hard for a horse to gauge height and distance on. This is because horses do not calculate depth perception as a human does, so if the jump has no height reference for the horse they may catch and trip. Most cross country jumps or fences have a diagonal bar, or are completely solid.
If the jump is excessively overgrown it is definitely worth checking by walking around it first as there can be hidden dangers in the over growth.
For ditches and the like it is worth checking, one how deep any water maybe, especially if it is dirty as you cannot tell how deep it is, and two how degraded the far side is. Horses can panic in a situation like this if they slip.
Most of the time a drop jump will be fine (imagine flat vertical drops cut into a small hill and jumping down them) but it's worth checking if the ground is slippery or waterlogged before descending.
Take into account the surroundings and anything that may cause your horse to be skittish or refuse the jump, you don't want to get thrown off into anything unpleasant or dangerous.
As for cross country jumping, this will depend. It is safe to assume if you're on a regularly maintained course then the jumps will have been maintained. If you're out on a hack and simply happen across one, I would follow the above and look for signs of degradation in the jump, and check the terrain before jumping.
Bear in mind... many jumps, fallen logs, fences etc that are not held in place by wings are often very solid! I once had a refusal at a telegraph pole jump and was dragged backwards over it smacking my jaw.