Hello - your Friendly Neighborhood Parachute Rigger here.
Sport chutes are pretty reliable, have been since the mid 90s to the point where more accidents happen under fully functional canopies.
If we disregard rank stupidity like tying the flaps shut1 or incompetent packing we find that the leading cause of malfunctions is bad body position on deployment. You are supposed to be face down, shoulders level before opening the thing. If you are on your back or tumbling you can have various parts wrap around you. High-performance canopies are somewhat more sensitive to the shoulders being level and can open in a decently fast spin. Depending on the spin direction and your altitude you may not be able to recover2.
The emergency chutes are built differently, emphasizing (obviously) reliability over being able to land at 100km/h. This means that twisted lines are either recoverable or not an issue.
If the jumper isn't paying attention to where he is, you may find that a leading cause of going splat is not using the perfectly good canopy on your back. Automatic activators have more-or-less eliminated this as a common cause3, but if the jumper thinks that the houses look a bit too big and deploys the main at the same time as the reserve opens they may decide to wrap each other up. Ram-air canopies are like badgers - they don't like each other very much and having two open at the same time is a rather delicate situation.
If we turn to landing accidents, we find that there are all kinds of ways to injure or kill yourself with a fully functional canopy overhead. The aforementioned 100km/h landing speed is no joke - hit an obstacle on the ground going that fast and it's exactly the same as jumping out of a car on the highway. Buildings, trees, power lines, vehicles, running propellers etc. have all claimed their sacrifices. And don't try to land on both sides of the fence at the same time. Or in the alligator farm across the road.
Mechanical failure of the system is very rare - riggers won't pack worn-out reserves, and if they are concerned about someone's main they will tell the dropzone management who will usually decline to provide services until it's fixed.
Many jumpers will tie various handles to the pack before travelling. Prevents the pack accidentally opening when curious security or customs pick it up by what looks like the carry handle. Remove before Flight!
Happened to me twice - the first time I looked up, then down, decided I didn't have enough time to fix it. The second time it was spinning the other way and I fixed it in 2 seconds.
Automatic activators fire at 750-1000 feet. If you are in freefall below that altitude you're usually done.
