I don't know about the US, but in the UK travel expenses can only be claimed against tax if you are travelling to a non-permanent workplace, i.e. if you work in an office and you cycle there every day then you can't claim, but if you were, say, a courier then you could claim the cost of maintaining your bike as allowable travel expenses and you could claim the cost of buying your bike and things like new wheels, tyres, etc. as capital expenses.