Serial nbrs on Ithaca 37's should fall into this pattern
1. 855,000 and beyond.
I spoke about that here:
http://thefiringline.com/forums/showthread.php?t=550432
This is at the change to the way the barrels attached to the frame. These are at the bottom right of the receiver, above the trigger guard. This is an immediate identification method that the gun is an interchangeable barrel model, keeping in mind that there are 2 3/4" receivers and 3" receivers. A 3" chambered barrel will fit and work fine on a 2 3/4" receiver but of course, can only chamber 2 3/4" shells.
you can identify the switch over to the 3" receivers by the word MAG- as either a suffix or as a prefix. most had it as a prefix. My early 3" 12ga says MAG-87xxxxxxxxxx
in time the word MAG was removed from the serial nbr. measuring the receiver is another way to determine which shell the gun was set up for. Ithaca added 1/4" to the receiver and selected components to allow the chambering/ejection of a 3" shell.
which leads into the fact that there are later production guns with "37" in the serial nbr, ie: M37xxxxxxx and guns with "87" in the serial nbr, ie: MAG-87xxxxxxxx,
the 87 is simply a year designator when the company resumed production (after a bancruptcy - or sale to a new owner, I would have to go read which, I simply don't remember). the 87 was subsequently changed back to 37.
there are some Model 38's. these were 37's with some short lived mods that simply didn't work out. the serial number will be 38xxxxxxxxxx. the barrel will also say Model 38
2. prior to 855,000
the serial nbr was on the front face of the receiver, as has been found and discussed. the receiver and barrel serial nbr "should" match each other. There are thoughts that there were a few that slipped out of the factory where the serial nbrs got out of sequence and then were caught, but the barrels were fitted to the receivers correctly, it was the nbr that was off.
depending on the year of the gun, the barrel serial nbr can have the choke designation as part of the nbr (with a dash) or the -nbr can be above or below the serial nbr.
I have Ithacas that are stamped both ways (part of, separate from)
Ithaca's "dash nbrs, -1, -2 and -4) are improved cylinder, mod and full, respectively. I have not seen nor read about a -3, but I could look into my book and see if it exists (I have the history of the Ithaca 37 as written by Walt Snyder - the official Ithaca historian, he also wrote a book on the rest of the guns and Ithaca itself)
on the older guns, a barrel could have been fitted to the receiver by a gunsmith, but other than the possible few that simply had the sequencing get out of order, having a barrel from one pre-855,000 gun fit another pre-855,000 is like hitting more than a couple numbers of the lottery. There always is a chance but the odds are slim due to the guns being spread around the country/world.
pre-855,000 barrels used a helical thread and having the lead for the threads on the receiver and the same on the barrel starting in exactly the same spot on more than one receiver/barrel set is possible, just not likely.