Does electricity have an associated entropy? One can certainly measure entropy changes associated with the generation of electricity, but does electricity itself have an associated entropy (maybe related to voltage, current etc.)?
 A: The energy of an electrical wave certainly undergoes energy loss through heat, so in that way is entropic.  This is a result of the material's resistivity through which the electricity is conducted.  The mechanism is primarily scattering of the energy through electron-phonon interactions, but electron-electron interactions do also occur.  In metals, the main sources of scattering are defects, including point defects (larger or smaller atoms within the crystal structure), grain boundaries and line defects (dislocations).
A: At the power station electricity is generated as work from a heat engine. Work is entropy free, so we have an entropy free electron-gas at the point of generation.
However, a thermodynamic gas will always equilibrate to the available degrees of freedom. In this case it is the electronic states of the conductor in the transmission wire. There will be a distribution of microstates that make up the observed macrostate of the electron-gas and this defines the entropy of electronic current.
As mentioned in the other answer irreversible processes cause energy loss and thus further increase the entropy (lower the chemical potential or voltage) of the electron gas. These process are proportional to the length of the conductor. However this is secondary. The electrical current does have an intrinsic entropy defined by the electronic states of the conductor.
In a theoretical conductor with only one state the electrical current would be entropy free.
A: Entropy is something related to thermodynamics, which is essentially a study about energy interactions. So you can use the concepts where ever you see an energy interaction.
Entropy is a measure of randomness resulting from the increase in heat energy supplied to a system. When you consider an electric circuit, you have an energy source which is the emf of the battery. There may be resistive components. One peculiar property of resistive components is that they develop heat. The principle behind it is simple. An electric resistor is a metal having a constant conductivity. A metal contains atoms tightly packed. So even though there are free electron availability in a metal, these electrons find themselves difficult to move through this less inter atomic spaces. The electrons on acquiring energy accelerates through this metal which increases the possibility of inelastic collisions with the atoms. So the electron loses some of its kinetic energy. This energy appears in the form of lattice heat. This is the origin of resistance.
This means that resistance correspond to energy lose of electrons , which in turn refers to the increase in lattice heat. This increase in lattice heat is irretrievable and is lost. By Joule's law we can see that the heat energy dissipated by  current I flowing through a resistor of resistance R is:
H=I^2*R
This energy in the crystal lattice induce thermal movement of electrons which will be in a random direction. (In the case of emf, it gives the electrons a proper direction to move, but that's not the case when you heat it). This randomness you can define by entropy. We define change in entropy by: 
dS=dQ/T, where dQ is the change in heat energy between two time intervals and T is the final temperature. Applying the corresponding quantities from the electric circuit analysis you could determine the entropy of the electrons inside the circuit.
