The idea is that the energy stored in a magnetic field is proportional to
where V is the volume of the magnetic field and B is the magnetic flux density (e.g. Teslas).
But the mass of a coil producing this magnetic field is proportional to

where l is the length of wire in the coil and
is the density of the wire — this scales (dependent on coil geometry) as an area instead of a volume. So, if you make the thing big enough, you could theoretically have any energy-to-mass ratio you wanted.
So we know that for a given magnetic field, the energy scales as length cubed and the mass scales something like length squared, the bigger this thing is, the better. So the real question is, how big does it need to be in order to do better than current methods (e.g. the space shuttle)?
Some equations:

For a single-layer air coil:
(r & l in millimeters)
For a brooks coil (maximum inductance per unit length of wire):

where a is the average radius of the coil, wound close-packed out of wire with a relatively small diameter. The cross-section of the close packed windings should be a square with an edge measuring 2a/3 on a side.

Taken from http://www.fnrf.science.cmu.ac.th/theory/magnets/
It is clear from the brooks coil that an optimal shape much more resembles a hoop than it does a single-turn solenoid.
Regardless of the structure chosen, obviously we want to have as strong a magnetic field as possible, corresponding to as much current (or as many turns) as possible, but there is a limit. Superconducting wire is limited by how much magnetic field it can be immersed in (colder=more field) and by how much hoop stress it can withstand. In this way, it is much like a “pressure vessel” with the magnetic field being the pressurised gas.
Since I am not a supercon magnet designer, I will have to extrapolate from existing magnets. These specs are kinda hard to come by, but as a data point www.cryoindustries.com listed a 15 Tesla magnet with 1.0 mm diameter wire carrying 300 amps having 33 kpsi hoop stress.
If we simplistically use this data for a large brooks coil, let’s see what we get. For example, 1 million turns of 1 mm diameter wire would make a square cross section 1 meter on a side. Making a brooks coil out of this would require a coil form with an average diameter of 1.5 meters. The inductance would be:

The energy storage would be:

Assuming supercon wire (mostly niobium) is about 9000 kg/m^3, the mass of this coil would be about

This gives us an energy density of about 1.3×10^6 J/kg. Compare that to the energy density of burning hydrogen + oxygen which gives us about 13×10^6 J/kg — so we are currently down by an order of magnitude in energy density, and even worse our coil is already enormous.
It is difficult to imagine realistically scaling up this coil by orders of magnitude. But let’s try anyway — if the coil were scaled to 10^7 turns, it would have 3.7×10^13 Joules and weigh 2.8×10^6 kg, giving an energy density of about 13×10^6 J/kg, on par with hydrogen + oxygen, and it would require over 6% of the world’s annual production of niobium (estimate taken from www.roskill.com).
The killer is that I think the assumption of 300 amps/mm^2 is not realistic in these coils. If we look at the magnetic field inside the first of these coils using

83 Tesla is far beyond the highest fields any superconducting material currently known can withstand at any reasonable temperature. I am sure the stresses inside such a magnet would also be very high.
You might be able to get away with a really large diameter loop of superconducting wire, but such a geometry does not seem very compatible with a normal “rocket.”
Perhaps if we are building a ship to travel to mars, superconducting magnetic energy storage could provide both protection from radiation (in the magnetic field) and energy needs.