We prove an inequality of the Loéve-Young type for the Riemann-Stieltjes integrals driven by irregular signals attaining their values in Banach spaces, and, as a result, we derive a new theorem on the existence of the Riemann-Stieltjes integrals driven by such signals. Also, for any pge1, we introduce the space of regulated signals f:[a,b]rightarrow W (a< b are real numbers, and W is a Banach space) that may be uniformly approximated with accuracy delta>0 by signals whose total variation is of order delta^{1-p} as deltarightarrow0+ and prove that they satisfy the assumptions of the theorem. Finally, we derive more exact, rate-independent characterisations of the irregularity of the integrals driven by such signals.