Let be the C∗‐algebra of quasicontinuous functions on and be the Cauchy singular integral operator, and let p ∈ (1, ∞). The paper is devoted to studying Mellin pseudodifferential operators with quasicontinuous ‐valued symbols on Lebesgue spaces , where is the Banach algebra of absolutely continuous functions of bounded total variation on the real line , and dμ(r) = dr/r. Applying obtained results on Mellin pseudodifferential operators with quasicontinuous ‐valued symbols, we study a Banach algebra of modified singular integral operators on the space , which is generated by the operators of the form where , , Rβ for is a singular integral operator with point singularities at 0 and ∞, Vα is the shift operator given by Vαf = f ∘ α, and α is a homeomorphism of [0, ∞] onto itself with . A Fredholm symbol calculus for the Banach algebra is constructed and a Fredholm criterion for the operators is established.