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the most common locations are -

underside of bottom bracket

lateral face of non-drive side dropout

front or side of head tube or front of head lug

side of seat tube, either just below seat lug or just above the bottom bracket

occasionally they can be found on the seat lug

sometimes the marking is struck so lightly it can almost be hidden by the bicycle's paint finish

best to search in the presence of strong illumination

there are some cycles which are manufactured without a serial number

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