All violent crimes and many economic crimes.

only the supreme court is authorised to give death penalty

Drug trafficking,murder,rape and lots more.

When organ supplies are depleted at hospitals

The Supreme People's Court must review and approve all death sentences.

This is an appellate review.

The death penalty is never issued by the Supreme Court, it is issued by the trial court.

I doubt rape leads to the death penalty.

Drug trafficking, weapon trading or smuggling, murder and corruption (when your best friends or political rivals take power and decide to screw you).

I read somewhere they were aiming to reduce and review the death penalty in China, but I seriously doubt it.