Burn it.
Full boxes.. ORIGINAL boxes? I would find a buyer for it.
Save it. While your ammo is probably completely safe to shoot (inspect cases first), it probably will have more value someday in the future. Or it would make a good conversation piece.
It should still shoot. I have some 1947 military 303 enfield that shoots fine. That said, it may be worth something to collectors, so check it out first.
There are ammunition collectors out there - I ended up with several boxes from a garage sale that were WWII manufacture from several different armories, some very obscure it turned out. I sold the group for about 100 times more than I paid.
You can do both but I recommend you check the value of the ammo before shooting.
I have several boxes of vintage ammo. I have kept a box of 30-30 Winchester Silver Tip because of the store and the price. The price was $4.95 and I bought it from a Toys R Us store.
That sounds like something rare.
Send it down line. As long as it is not corroded it will be fine. That’s my experience at least. Before a tragic boating accident I sent lots of old ammo down range.
Here is some advice form CMP (Civilian Marksmanship Program)
http://thecmp.org/cmp_sales/ammo-parts-accessories/warning-surplus-ammunition/
I even have various full boxes of unused Wanda shotshells. Not counting unopened 550rd spams cans of steel core .308 and 30-06 from dad's new old stock piles. I think some of the stuff is actually worth some money now, but probably not enough to make it worth hanging on to all these years...
I just found a few boxes of ammo I bought 40 years ago! Still shoots.
Dominion 30-30
9MM Remington
.44 Mag.
I even found a box of 44-40 Winchester for a rifle I sold 35 years ago!
I reloaded lots of .38 cal into IVI military cases I picked up back in 1968! I noticed the primer hole was off centered. Brok several depriming pins.
I found some 22 shorts my dad had from the late 1940s and out of 60 or so rounds not a single dud. I did leave some alone for posterity.
As long as the commercial ammo had been stored high and dry and not in an overly hot place is will be just fine to shoot. The GI 38 is a bit iffy since you have no way of knowing about storage and I have had misfires with similar stuff. None of it is “collectable” except that you might find someone who collects military markings for the 38. I have samples of all these and they work just fine as needed for test firing etc. Check boxes for signs of moisture and the primers and bullet cannelures for signs of fuzzieness.
Shoot it! BUT BE CAREFUL, if any rounds just don’t “look” right, DO NOT SHOOT THAT ROUND!
Have your Mother-In-Law try it first.
The Super Police was popular at a time when hollow point bullets were relatively undeveloped, and could not be counted on for consistent performance. The slow, long Super Police bullet would yaw on striking tissue, creating a larger wound channel.
Here's what it looks like.
If you have a full box, that would be of interest to collectors.
The packaging appears to be in excellent conditions which implies that the ammo has been stored in reasonable conditions. Moisture, excessive heat and corrosive atmospheres are the main enemies of ammunition.
As other posters have noted, the visual condition of the cartridges offer the best clues to whether it is still safe to shoot.
I personally wouldn't hesitate to shoot the ammunition for recreational purposes. Ammunition used for serious purposes should be of current vintage.
Changes in atmospheric pressure will cause moisture to enter a cartridge that may lead to a dude or even a squib that may leave a bullet in the barrel. Not so much of a problem for rimfire cartridges. For storage, keep ammo in a sealed plastic bag that can expand and contract with pressure. Not a bad idea to seal the primers with a dab of fingernail polish. Some WWII stuff still had corrosive primers.