I will give you my personal mental image of unpolarized light, maybe it
will help.
In a given point in space, the E field is a vector lying in the
plane perpendicular to the propagation. In this plane, if you put the
tail of the vector at the origin, then the tip of the vector is a point
jerking in a random fashion around the origin. The important thing is
that it is random, not periodic, as purely monochromatic light
cannot be unpolarized.
If the light is narrow-band, the movement will look kind of periodic
(and thus polarized) over short time scales. You would then be able to
define an “instantaneous polarization”. But this polarization will
slowly change over the time scale corresponding to the bandwidth. You
cannot assume anything about the instantaneous polarization: it could be
linear, elliptical or circular. I would assume though that it changes
continuously, unless the spectrum of the light is quite heavy-tailed:
discontinuities in the time domain always make heavy tails in the
frequency domain.
If it is white light, then the tip of the vector is just jerking
randomly, with a hardly discernible frequency corresponding to the
middle of the band. Maybe more a time scale than an actual frequency. It
would be very hard to identify an instantaneous polarization, because
such polarization would be changing practically in the same time scale.
You could describe both situations as the superposition of two fields
with perpendicular polarizations: the combined polarization can be
computed from the amplitudes and phases of the components. But since
those amplitudes and phases have a finite coherence time, then your
polarization is always changing.