If you have a piece of wire that is a superconductor (say Niobium), and you hold the temperature at one end above $T_c$, and the other end below $T_c$ -- would you have a state of both superconducting / normal? Would this be gradual across the wire, or a sharp point?
It is unfortunately a rather difficult question to answer because 1D superconducting systems, especially close to the critical temperature, will exhibit quantum phase slips. Those are giant waves of the phase of the superconducting order of magnitude propagating over the wire, and can be thought as a tiny normal region propagating inside a superconducting region. It is in a sense the 1D version of the vortex, except phase slips do not require magnetic field to be applied to the superconducting system.
In short, in 1D and close to the critical temperature, you will have competition between superconducting and normal orders, and this competition is rather random in space.
In 2D and 3D, you would have droplets of superconductivity inside normal regions (still in the form of fluctuations and competitions between the two phases), up to the complete installation of either the superconducting or the normal phase. Check nucleation entry on Wikipedia to understand a bit more about that kind of competition.
My take on the question is a little bit different than FraSchelle's. I consider only the 3D case. By holding the temperature fixed at both ends of the wire, one below $T_c$ and one above $T_c$, I'd say that locally the stable phase is the superconducting phase in the region where the temperature is below $T_c$ and in the other region the stable phase would be the non superconductor one. As such, macroscopically at least, since the regions where $T<T_c$ and $T>T_c$ are well defined, I'd expect a spatially abrupt change in the wire's electrical resistivity: from zero in the superconducting region to a non zero value (that is a function of a spatial coordinate along the wire, i.e. along the temperature gradient) in the other region. Just like an interface ice/water in equilibrium. What happens microscopically is related to nucleation as FraSchelle mentioned and I'd expect fluctuations where very tiny regions switch between the superconducting and non superconducting state.
On the other hand if you pass current in this wire, things get more complicated. There would be a Joule heating in the non superconducting part modifying the temperature profile, hence provoking the motion of the superconducting/non superconducting interphase toward the colder end. There would also be a Thomson effect, albeit much smaller than the Joule effect, either cooling or heating the wire depending upon the sign of the Thomson coefficient for the wire's material. I think this latter effect would take part in the whole wire, unlike the Joule effect.
Essentially, you are describing a common device for measuring the level of liquid helium in cryogenic containers (available in the commerce).
You take a long superconducting wire. The portion inside the liquid helium is superconducting and the portion outside is in the normal state. A measurement of the wire resistance gives you the length that is in the normal state.
This measurement will put a small current through the wire and this current, thought Joule heating, will slightly change the wire-fraction between superconducting and normal states.
So, the answer to your question is: part of the wire will be superconducting and part of the wire will not. The boundary is a reasonably sharp point.