Normally, for Normal Trap Cards and Counter Trap Cards, after they are activated and resolve they are destroyed and sent to the Graveyard. Continuous Trap Cards with the "Continuous" icon and Trap Cards like "Metalmorph" and "Blast with Chain" which equip themselves to monsters remain face-up on the field after activation.
These TCG rulings were issued by Upper Deck Entertainment and have since been deemed unofficial by Konami. They were previously considered official and can still be correct, as long as they follow the official gameplay rules, or unless Konami has issued a ruling that says otherwise. References: 
↑Konami Judge Program Rulings are considered unverified, as their source(s) are not publicly viewable. They are not confirmed as applicable to the TCG until they have been officially published by Konami.