It depends on who hit the shuttle in which situation. According to the Laws of badminton:
§220.127.116.11 It shall be a let if, after the service is returned, the shuttle after passing over the net is caught in the net;
So if the racket side hit the shuttle during the rally, it's a let, i.e. you replay the rally.
In all other cases, it's a fault of the side which hit the shuttle:
§13.2.2 It shall be a fault if, in service, the shuttle after passing over the net, is caught in the net.
§13.3.2 It shall be a fault if, in play, the shuttle fails to pass over the net;
While the shuttle is in flight, the latter rule does not apply. Per §15.1, only if the shuttle is caught in the net and starts to drop down, the fault is called.
So much for the shuttle side. When the racket touches the net, it's a fault as well:
§13.4.1 It shall be a fault if, in play, a player touches the net (..) with racket (..).
Assuming the shuttle side hit the shuttle last, you have to evaluate what happened first - per §15.3, the first fault ends the rally.
- If the racket touched the net before the shuttle hit the net, it's a fault of the racket side and the point goes to the shuttle side.
- If the shuttle hit the net and started falling downwards (even a little bit of tumbling) before the racket hit the net, then it's a fault of the shuttle side and the point goes to the racket side.
- If the shuttle got caught in the net just as the racket hit the net and you can't decide which happened first, it's a let, per §14.2.7.