The class 37s used on duties on the West Highland and Far North lines were based at Eastfield and Inverness depots respectively following their introduction in 1981. The following information is taken from "Scottish Class 37s Volume One" by Nick Meskell:
Eastfield depot
In the summer of 1982, Eastfield depot had an allocation of 31 class 37s, which had been primarily reallocated from East Anglian depots at Stratford and March. All of those listed below were fitted with air and vacuum brakes as well as steam heat boilers, unless otherwise stated:
37012 "Loch Rannoch"
37014
37018 (Air and vacuum brakes, no train heat)
37022
37026 "Loch Awe"
37027 "Loch Eil"
37028
37033
37037
37039
37043 "Loch Lomond"
37051
37081 "Loch Long"
37085
37108
37111 "Loch Eil Outward Bound" (Named 1985)
37112
37117 (Vacuum braked only, no train heat)
37144 (No train heat)
37147 (No train heat)
37149 (No train heat)
37157 (No train heat)
37179 (Steam heat boiler isolated)
37184
37188 "Jimmy Shand" (Named 1985)
37190
37191 "International Youth Year 1985" (Named 1985)
37192
37196 (Vacuum braked only, no train heat)
37237 (No train heat)
37253 (Vacuum braked only, no train heat)
By the summer of 1984, Eastfield depot had gained a further thirteen class 37s, bringing the total allocation to 44, to replace fifteen class 25s on freight duties:
37011 (Transferred from Inverness)
37013 (Vacuum braked only, no train heat)
37090 (Vacuum braked only)
37124 (No train heat)
37148 (No train heat)
37150 (No train heat)
37151 (No train heat)
37152 (No train heat)
37171 (No train heat)
37172 (No train heat)
37175
37178
37263
37264 (No train heat)
Inverness depot
Inverness depot had a small allocation of nine lass 37s, for use exclusively on the Far North lines. This was the situation in summer 1982, although as with Eastfield the allocation remained fairly static until the 37/4s began to come in during 1986:
37011 (Vacuum brakes, steam heat boiler, subsequently moved to Eastfield)
37017 (Vacuum brakes, steam heat boiler)
37025 (Vacuum brakes, steam heat boiler)
37035 (Vacuum brakes, steam heat boiler)
37114 "Dunrobin Castle" (Vacuum brakes, steam heat boiler, named 1985)
37183 (Air and vacuum brakes, steam heat boiler)
37260 "Radio Highland" (Air and vacuum brakes, steam heat boiler)
37261 "Caithness" (Air and vacuum brakes, steam heat boiler)
37262 "Dounreay" (Air and vacuum brakes, steam heat boiler)
--- old post above --- --- new post below ---
Otherwise, my vague recollection is that most if not all Scottish 37s were based at Motherwell until they appeared in quantity at Eastfield and Inverness around 1981 to replace 27s and 26s on the West Highland and North lines.
The details I have pre-1981 are sketchy, but there appears to have been a batch delivered new to South Wales in 1963, D6837 - D6857 and later 37137 - 157, which subsequently moved to Scotland in August/September 1966 and were allocated to Haymarket and Polmadie depots, before being settled at Eastfield in 1972. Some remained there until late 1979/early 1980, when they were reallocated to Motherwell.
For reference, 37056/117/125/133/137/139/145/150/152/154 - 156 are shown as being allocated to Motherwell depot at the end of 1982. Evidently there's a bit of crossover with those Eastfield locos listed above.