In Missouri the title *is* the registration, and I keep mine in my safe deposit box. What we have on the window is a safety inspection sticker.
What I keep *in* the car is a photocopy of the license tax receipt, with the details of the address blacked out (house number and zip code). My street is several miles long, so just knowing what street it is wouldn't help a thief. No one ever drives my car except me or (very rarely) DH, so that isn't an issue.
It's been at least a decade since I've been stopped for any reason, but my recollection is that officers here don't ask for registration, just license and proof of insurance. They will call in to check the status of the tags and your SSN, and as long as the vehicle isn't reported stolen, they won't question your right to be in possession of it. (Well, I guess maybe if you had a long list of priors for GTA, they might question.)
Now I'm wondering if I should photocopy my title and carry that in the car in case I drive out of state, in case some officer asks me for it.
One funny thing *did* happen to me about 15 years ago when I was stopped for speeding. They ran my SSN and discovered that there was a bench warrant out from Louisiana, where I had previously lived. I was ordered to contact the court and report in. When I called, it turned out that the warrant was issued because I had not renewed my car insurance policy on the car I had when I lived there -- because I had moved out of state and bought a new policy when I changed my domicile. It turns out that the law in Louisiana is that if you let an insurance policy just lapse without notifying the insurer that you no longer own the vehicle or have taken out a policy elsewhere, it generates an automatic bench warrant. Since I no longer lived there *AND* I had sold the car about 7 years previously, they decided to vacate the warrant.