@Daniel_Mathias gave a very helpful link which has all the 12x5 solutions in a text file. So some simple code allows us to see that of the 1010 12x5 solutions, there are 264 with 1 straight cut. But, sadly, none with 2 or more cuts. A few examples of the former are:
12
FFPPP IIIIINN
LFFPP ZZWNNNT
LFXUU VZWWTTT
LXXXU VZZWWYT
LLXUU VVVYYYY
81
FYYYYWZZUUU I
FFFYWWZTUXU I
PFVWWZZTXXX I
PPVNNNTTTXL I
PPVVVNNLLLL I
464
I LLLLUUXYYYY
I VVVLUXXXYPP
I VZZFUUXTWPP
I VZFFFNNTWWP
I ZZFNNNTTTWW
Edit: Now that I've understood @DanielMathias's point more thoroughly (see discussion under his answer), the question becomes "how few straight cuts of any length can be done". For 12x5, there are 18 with only 31 internal cuts (which is the fewest cuts for any 12x5). Examples include:
12
FFPPPIIIIINN
LFFPPZZWNNNT
LFXUUVZWWTTT
LXXXUVZZWWYT
LLXUUVVVYYYY
60
FLLLLIIIIINN
FFFXLZZWNNNT
PFXXXVZWWTTT
PPUXUVZZWWYT
PPUUUVVVYYYY
570
IPPPNNNWWVVV
IPPNNZZXWWTV
IUUFFZXXXWTV
IUFFZZYXLTTT
IUUFYYYYLLLL
623
IVVVWWNNNPPP
IVZZXWWTNNPP
IVZXXXWTFFUU
IZZYXLTTTFFU
IYYYYLLLLFUU
747
LLLLNNNFFPPP
LIIIIINNFFPP
VZYYYYWTFXUU
VZZZYWWTXXXU
VVVZWWTTTXUU
Further edit:
If we are just looking to minimize the number of cuts, then we can consider other lengths. I used the 3x20, 4x15, 5x12, and 6x10 solutions from the pentomino website and found that there are 9 solutions that have only 30 cuts. They are:
308
IIIIINLLLL
PPPFFNTTTL
PPFFWNNTZZ
UUXFWWNTZV
UXXXYWWZZV
UUXYYYYVVV
609
IIIIIWWVVV
LLLLWWXUUV
LNNNWXXXUV
PZZNNFXUUT
PPZFFFYTTT
PPZZFYYYYT
1089
IPPYYYYVVV
IPPXYFFUUV
IPXXXWFFUV
IZZXWWFUUT
IZLWWNNTTT
ZZLLLLNNNT
1324
IVVVLLLLPP
IVWWTTTLPP
IVFWWTZZXP
IFFFWTZXXX
IFNNYZZUXU
NNNYYYYUUU
1419
IWWTTTLLLL
IYWWTFNNNL
IYYWTFFFNN
IYVPZZFXUU
IYVPPZXXXU
VVVPPZZXUU
1653
LLLLIIIIIT
LFZZPPPTTT
FFFZPPXUUT
FWWZZXXXUV
WWNNNYXUUV
WNNYYYYVVV
1654
LLLLIIIIIT
LFZZPPPTTT
FFFZPPXUUT
FWWZZXXXUV
WWYNNNXUUV
WYYYYNNVVV
1963
LPPYYYYVVV
LPPFFYXUUV
LPFFWXXXUV
LLZFWWXUUT
ZZZNNWWTTT
ZNNNIIIIIT
1964
LPPYYYYVVV
LPPXYFFUUV
LPXXXWFFUV
LLZXWWFUUT
ZZZWWNNTTT
ZIIIIINNNT