I've used them all, they each have their place, based upon your intended purpose at the moment, as well as being limited by your budget.
After going down the rabbit hole and buying, installing and testing a half-dozen of the most popular under-saddle, and another half-dozen of the most popular surface transducers...as well as building my own from a wide variety of materials, including PVDF piezo film, piezo discs, piezo cable, and SBT piezo rods, and I've come to a few conclusions:
1. Surface transducers produce a more 'woody' sound (maybe more 'natural' sound), at the expense of also picking up ALL scratchy/surface noise of the instrument and being easily prone to feedback at higher volume levels - as such, amp/speaker placement relative to uke placement is critical to minimize feedback when playing live.
2. Under-saddle transducers (UST) are not prone to feedback, even at higher volume levels, and produce a slightly compressed, but very articulate sound, very crisp, but can tend to sound 'plasticky' depending upon what strings you use, and what material your saddle is made from. Plastic saddles have the worst of the 'plasticky' sound, bone saddles have less, and the least 'plasticky' sound I've found is from using either a rosewood or ebony saddle, which also tends to be WARMER, and more 'WOODY' than the others. USTs usually require a preamp, either installed in the uke or external because their output voltage tends to be lower than surface transducers. Most cheap asian rod piezo pickups have enough output to not need a preamp to get the level up, but can benefit from one any way to tame the 'piezo quack' that remains even with this type of pickup. The flex-cable (Like the LR Baggs Five-O, Mi-Si Acoustic Trio) and PVDF piezo film (B-Band calls theirs emFit) definitely needs a preamp as the output voltage is about 1/4 of that of the rod piezo.
3. A mic is a good way to perform live if you have a PA system already set up, and it's not something you have to worry about if there is a sound engineer already running the FOH mixer, however, you pretty much have no control over your sound, and cannot move around at all, since you need to have the mic right in front of your uke in order to be heard through the speakers.
4. For recording, having tried all of the above, as well as some very fancy/expensive studio condenser microphones, and I've found that for recording purposes, there is nothing that is going to give a sound as close to your own ears as a good micrphone, or pair of microphones, since they are designed more to emulate the human hearing. I am about to sell off most of my studio gear since I purchased an Apogee MiC, which works with Mac, iOS and Linux out of the box with no drivers (there are 3rd-party drivers for Windows, but I have not tried them) and sounds AMAZING.
Better than my AKG studio mics. All it needs is a single cable into your device. There is a HUGE benefit to this simplicity as opposed to the octopus spaghetti nightmare of cables, a matched pair of stereo condenser mics, an overhead mic, and a 'room mic' into various preamps, setting and checking levels for untiy gain, running it all into your recording interface and then doing a few test recordings... and then finally you can record -- if you can even remember the music that inspired you to capture something 30 minutes ago now...not.
I no longer have a dedicated space for a 'studio' setup, so the Apogee MiC is ONE cable, and ONE mic stand/mount, and I've been ready to record in 5 mins or less, and it's much easier to get reproducible results since you dont have to tweak dozens of knobs to set everything up each time.
Simply put, for recording, there is NOTHING that I've tested for pickups that give me back the acoustic sound THAT I HEAR, when playing the uke, other than a good microphone(s) that are positioned properly.
The Apogee MiC's recorded sound varies based upon placement and distance relative to your uke, and I have several software plugins that give me a virtual stereo sound (or even surround-sound), using principles of psychoacoustics.
This means to get a wider, and bigger, cleaner and more natural sound without using lots of compression and artificial reverb, I can use just one mic, and these software plugins as needed.
I do not have a full song recorded yet, just some short tests, and so far this is the closest thing to what I hear with my own ears...(nothing to share at this time-sorry)
I'll be using this method to record the songs I've written on the ukulele over the past 2 yrs - STAY TUNED!