A new muon beam facility, called the Experimental Muon Source (EMuS), was proposed for construction at the China Spallation Neutron Source (CSNS). The design of the complex muon beamlines for the EMuS baseline scheme, which is based on superconducting solenoids, superferric dipoles and room-temperature magnets, is presented herein. Various muon beams, including surface muons, decay muons and low energy muons, have been developed for multipurpose applications. The optics design and simulation results of the trunk beamline and branch beamlines are presented. With a proton beam power of 25 kW at a standalone target station that consists of a conical graphite target and high-field superconducting solenoids, the muon beam intensity in the trunk beamline varies from 107/s for surface muons to 1010/s for high-momentum decay muons. And at the endstations, these values vary from 105/s for surface muons to 108/s for decay muons.