It is well known that vanilla fixed for floating swaps usually have a bit of gamma, but does a floating for floating (basis) swap have any? For the sake of simplicity, let's assume that both legs of the swaps are in the same currency.
In practice a 3s-1s basis swap has negligible gamma. Imagine putting on the basis swap, then the basis swap market moves. The resulting profit or loss is the present value of a fixed annuity, whose value depends on outright rates, not basis swaps. Sometimes there could be a correlation between rates and basis swaps, which could make this covariance feel like a gamma position, but this is relatively rare in my experience.