swaps - Discount curve and payment frequency

In case of uncollateralised trades, where we use LIBOR rates for discounting, does the LIBOR tenor have to match with the payment frequency?

For example, one of the swap leg pays USD floating amount every 3 months, does this suggest that we should use 3M USD Libor for discounting? (and 1M USD LIBOR for monthly payment etc.)

If this is true, what if the payment frequency is once every 3 years. What would be the best LIBOR tenor to use for discounting?

If not, is there any market convention of choosing the suitable default LIBOR curve for different currencies, irrespective of the payment frequency?

Thanks.