In this paper, we mainly simulate the characteristics of the ground state dark soliton and the multipole dark soliton in the nonlocal and cubic-quintic nonlinear medium. Firstly, the influences of the degree of nonlocality on the amplitude and the width of the dark soliton in the self-defocusing cubic-and self-focusing quantic-nonlinear medium are studied. Secondly, we find the nonlinear parameters affecting the amplitude values of solitons, but the refractive index induced by the light beam is always a fixed value. The numerical results show that the ground state dark soliton can be propagated stably alone the z axis, and the stable states of the dipole soliton and the dark tri-pole and quadru-pole solitons are stable. However, some quadru-pole dark soliton is unstable after propagating the remote distance. Furthermore, we also discuss the characteristics of the ground state dark soliton and the dark dipole soliton in the local cubic-nonlinear and nonlocal quantic nonlinear media. Both the amplitude and the beam width of the dark ground state soliton and dark dipole soliton are also affected by the degree of nonlocality and nonlinearity. Two boundary values of the induced refractive index change with the variations of the three nonlinear parameters. The dark soliton and the dipole dark soliton are more stable in the self-focusing cubic nonlinear and the nonlocal self-defocusing quantic nonlinear medium than those in the self defocusing cubic nonlinear and nonlocal self-focusing quantic nonlinear medium. The powers of single dark soliton and dark tri-pole soliton decrease monotonically with the increase of propagation constant when the cubic-quintic nonlinearities are certain values and these degrees of nonlocalities are taken different values. Furthermore, we also analyze linear stabilities of various nonlocal spatial dark solitons. And we find that the dipole dark soliton is unstable when the propagation constant is in the region-0.9,-1.0. These properties of linear stabilities of other multi-pole dark solitons are the same as their propagation properties.