What can and cannot be planted on grave sites?

It depends on the cemetary. If it is one connected with a funeral home, call them and ask. Some have rules as to size and type of headstone and flowers that can be left at the gravesite. If it is a public cemetery and you want to plant something I would keep it small. You don't want something that will spread to other graves or get big and bushy or require a lot of care. Some cemeteries have grounds keepers who mow and keep the cemetery neat. You should check on this. What ever you plant might just get mowed down. To me just a thick carpet of grass is very restful.