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: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are positive constants. Among others, we prove there are [Formula: see text] and [Formula: see text] ([Formula: see text]) such that the constant solution [Formula: see text] of system is locally stable when [Formula: see text] and is unstable when [Formula: see text], and under some generic condition, for each [Formula: see text], a (local) branch of nonconstant stationary solutions of system bifurcates from [Formula: see text] when [Formula: see text] passes through [Formula: see text], and global extension of the local bifurcation branch is obtained. We also prove that any sequence of nonconstant positive stationary solutions [Formula: see text] of system with [Formula: see text] develops spikes at any [Formula: see text] satisfying [Formula: see text]. Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from [Formula: see text] when [Formula: see text] passes through [Formula: see text] can be extended to [Formula: see text] and the stationary solutions on this global bifurcation extension are locally stable when [Formula: see text] and develop spikes as [Formula: see text].