  Transmission matrix measurements of multimode fibers are now routinely
performed in numerous labs, enabling control of the electric field at the
distal end of the fiber and paving the way for the potential application to
ultrathin medical endoscopes with high resolution. However, the process of
building an experimental setup and developing the supporting code to measure
the fiber's transmission matrix remains challenging with full details on
experimental design, data collection, and supporting algorithms spread over
multiple papers or lacking in detail. Here, we outline a complete and
self-contained description of the experiment we use to measure fully
polarization-resolved transmission matrices. Our specific implementation of the
full polarization experiment is new and is easy to align while providing
flexibility to switch between full-polarization and scalar measurements if
desired. We introduce a new method to measure and account for the phase and
amplitude drift during the measurement using a Levenberg-Marquardt nonlinear
fitting algorithm. Finally, we describe creating distal images through the
multimode fiber using phase-to-amplitude shaping techniques to construct the
correct input electric field and show that results are insensitive to this
choice as quantified by measuring the contrast of a razor blade at the distal
end of the fiber. Throughout the paper, we discuss applications of our setup
and measurement process to a variety of specialty fibers, including fibers with
harsh environment coatings, coreless fibers, rectangular core fibers, pedestal
fibers, and a pump-signal combiner based on a tapered fiber bundle.
