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In this paper, we study stability, bifurcation and spikes of positive stationary solutions of the following parabolic–elliptic chemotaxis system with singular sensitivity and logistic source: ut = uxx − χ u vvxx + u(a − bu), 0 < x < L,t > 0,0 = vxx − μv + νu, 0 < x < L,t > 0,ux(t, 0) = ux(t,L) = vx(t, 0) = vx(t,L) = 0t > 0, where χ, a, b, μ, ν are positive constants. Among others, we prove there are χ∗ > 0 and {χk∗}⊂ [χ∗,∞) (χ∗∈{χ k∗}) such that the constant solution (a b , ν μ a b ) of system is locally stable when 0 < χ < χ∗ and is unstable when χ > χ∗, and under some generic condition, for each k ≥ 1, a (local) branch of nonconstant stationary solutions of system bifurcates from (a b , ν μ a b ) when χ passes through χk∗, and global extension of the local bifurcation branch is obtained. We also prove that any sequence of nonconstant positive stationary solutions {(u(⋅; χn),v(⋅; χn))} of system with χ = χn(→∞) develops spikes at any x∗ satisfying lim infn→∞u(x∗; χ n) > a b. Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from (a b , ν μ a b ) when χ passes through χ∗ can be extended to χ = ∞ and the stationary solutions on this global bifurcation extension are locally stable when χ ≫ 1 and develop spikes as χ →∞. [ABSTRACT FROM AUTHOR]