For an angel or demon to truly die their essence has to be consumed. This means that once their physical form has been defeated (vessel or no) the energy that makes up their being lingers. Other supernaturals can consume this power to make them stronger, but most angels do not try to consume demon essence.
To break it down, the fallen after a battle between angels and demons linger in the remains of their physical forms when they are not using a vessel. When they use a vessel and the vessel is killed, they automatically return to their plane of origin. When not using a vessel, a higher level angel or demon has to go among the fallen and shepherd the essence to heaven or hell to recuperate. If they are not returned to their plane by another of their faction, their very essence will eventually dissolve and they will truly die. Restored angels/demons suffer a power loss and often times at least partial memory loss, in addition to taking a substantial amount of time to get back into fighting shape.
Then there is the quick way to die. Angels, demons, and other supernaturals are capable of consuming the essence of an angel or demon in battle, adding to their own strength. In battle, after killing the physical form, they simply absorb the essence of the fallen, truly killing them.
Hope this makes sense! (Let me know if it doesn’t… sometimes my head canons are a lot more difficult to explain than I think)
So tattoos so far: Angel wings on the back. Enochian words (what exactly they are going to say is yet to be determined…) likely on the spine and/or inside of the arms.
Because your character is a cop (at least for a time), the tattoos need to be easily covered by a shirt, hence being a little leery of wrist/neck tattoos.