Is state variable analysis used much in circuit analysis these days?
I assume that by the State Variable Formulation you mean 
dx/dt= Ax + Bu, 
where A is the state matrix and B is the input matrix.

The State Variable Formulation is not popular with programmers. For better or worse, a more popular circuit analysis method today is Modified Nodal Analysis, which is used in simulators.

The difference between these two approaches are confined to the details of how the matrix math is written. For lumped, linear elements, they all analyze the same state variables of the circuit. When these solvers work, they provide the same answers. The challenge is to write the matrix in a way that:


Is not too difficult to understand and create
Can be fed to a solver that provides useful solutions