When the Marlin model 60 was introduced in 1960, there was no law requiring manufacturers to put serial numbers on their guns. Many did anyway, for record keeping and warranty repairs.
It was mostly the inexpensive guns like 22s and shotguns that didn't get the serial number. That all changed in 1968 with the passage of the '68 Gun Control Act.
Marlin did not serialize the model 60 until 1968, so if yours is older than that, you can check the date code on the barrel. The date code is located just ahead of the receiver on the left side of the rifle. This only applies to guns made between 1962 and 1968, if you have no serial number and no date code, you gun was made in 1960 or 61.
Of course all of these methods assume the parts on the gun are original.
The first letter of the barrel date code indicates the month, see the chart below:
A = Jan
B = Feb
C = March
D = April
E = May
The second letter indicates the year, again they didn't use this system until 1962
F = 1962
G = 1963
H = 1964
I = 1965
J = 1966
K = 1967
So if your barrel code is HH your gun was made in August of 1964
Starting in 1968 Marlin used the first two digits of the serial number to indicate the year the gun was made. So a gun with serial number 703778 was made in 1970.
Then in 1973 they changed the first two digits to be subtracted from 100 indicated the year of manufacture. So a gun with serial number 210566 was made in 1979 (100 - 21 = 79).
When they got to the year 2000, they used two zeros eg: 0056772.
Beginning in 2001 the first two digits were now subtracted from 2100. Serial number 980122 was made in 2002 (2100 - 98 = 2002)
Then in 2010 they began using the MM prefix for standard models and the DLX/DL prefix for the deluxe models. The serial numbers are now sequential.
There are other ways to identify the year a model 60 was made based on the features.
If you have the older style scooped charging handle, your gun was made before 2005
Conversely if you have the newer style "spindle" charging handle, you rifle was made after 2005
If your rifle has a factory 19" long barrel, it was made after 1999
If your magazine holds 15 rounds and is shorter than the barrel (see picture below) your rifle was made between 1989 and 1999
If your barrel is 22" long and the magazine is full length, your gun was made before 1989.
If your gun has the Glenfield name, your gun was made before 1983
If your gun has the last shot hold open/bolt release lever (just ahead of the trigger guard in the picture below), it was made after 1984
If your gun has a checkered stock, is not a deluxe model, and it came from the factory that way, it was probably made before 1983
If your gun has two rear trigger guard screws your gun was made before 1981
If your gun has an aluminum trigger guard it was made before 1977 (see picture above)
If your gun has the front sight and magazine band combined in one unit, your gun was made before 1975
If your rifle has no serial numbers, it was made before 1968