In chron order:
11358 - 14 years
10027 - 2+ years
07054 - 2 years
11201 - 3 years
21208 - 1 year
21201 - 1 year
11358 - 9+ years
(The first 5 years of my life I lived in 10457, but I had to look that one up.)
48198
? St. Charles, MO
63366
? White Lake, MI
30603
30605
30603
63366
? Breckenridge Hills, MO
30303
30605
30606
30308
43209
Yes, I moved around in Athens, GA often.
Yes, because I’m pathetic, I can list them all* quite quickly.
33830
33841
33880
*From my school years on, anyway.
And as I read the OP again (properly), I have to admit that of those zip-codes I only know four by heart.
In chronological order, not counting the hospital where I was born:
60913
10509
70062
70123
? (Great Lakes RTC)
70123
22206
22304
22201
22206
22041
22201
22204
22201
22204
22205
22204
General Electric’s headquarters in Schenectady.
Zip codes:
43017
43235
Postcodes:
2300
2303
and those that I could only find by checking:
2031
2304
2322
In order:
46360
49120
48021
46526
47906
46703
47421
46226
46256
46250
46219
23666
23669
27936
Since Nov 2005.
96817
32819
32207
90019
96826
89103
96825
I don’t know the one where I lived until I was 5, but then again, I don’t think the ZIP Code had even been invented yet. There were “zones”, like Brooklyn 4, NY, but I don’t even know THAT code. I do remember the phone number, though, if that counts 
The rest of them, no problemo:
11804
02881
02879
02881
11804
07017
07076
08876
08844
There haven’t been many:
67801 (Dodge City, KS)
66502 (Manhattan, KS)
66212 (Overland Park, KS)
64112 (Kansas City, MO)
64114 (Kansas City, MO)
44221 (Cuy. Falls, OH – I have a vague recollection of the house)
44305 (Goodyear Heights, Akron, OH)
15213 (Carnegie Mellon Univ., Pittsburgh, PA)
15217 (Squirrel Hill/Greenfield, Pittsburgh, PA)
15232 (Shadyside/East Liberty, Pittsburgh, PA)
20121 (Centreville, VA)
Assuming, of course, that I got them all right. Adding the locations doesn’t bother me too much, since they’re easy to find online anyway.
Chronologically:
21114
[don’t remember]
[don’t remember]
14853
14850
21032
21410
21xxx*
21054
89523
89503
89509
*I I know I moved between 21410 and 21054 time period, but I can’t remember the ZIP code.
That’s my current ZIP code!
90210 (BHPO but not BH)* - House I grew up in, only one my parents ever owned
90293 - On campus at UCSD
90239
90025
90232 (I had to look this up, I only lived here about six months)
90025 (again)
also one year in Germany, at 3400.
*Most of 90210 is actually within the city limits of Los Angeles but served by the Beverly Hills PO.
90045
90278
92110
92111
92108
92109
92126
92107
92131
92123
92117
All those 92 zips are different parts of San Diego county - I moved around a lot during and right after college.
Lemme see, from birth . . .
94301
94306
90024
80218
80209
94115
94403
94062
Huh, I guess I haven’t moved around all that much.
77006
78705
78721
77063
77515
77042
and various foreign postal codes that I don’t remember
I can only remember the ones where I spent a substantial amount of time:
91331 Pacoima.
95616 Davis.
93117 Isla Vista.
and my supersecret current zip code.
I should remember Palo Alto. It looks like 94306.
It is.
Before zip codes: Brooklyn 12
11412
11423
23226
19901
19806 (I think)
30274
30236