How to figure out the type of the bifurcation in a dynamical system?

Suppose we have a dynamical system

$dot{x} = f(x,r)$

in which x is a state variable and r is a bifurcation parameter. How to figure out which kind of bifurcation(s) (e.g. saddle-node, transcritical, pitchfork, hopf and etc) the system undergoes?

Edit 1: consider the space as 1D or 2D.