I own many microphones for my saxophones, for live and recording use. Most of my performances do have sound systems in use by my band when we play, yet I don't use a microphone for the sax all the time. The saxophone can actually be too loud in some rooms, all by itself.
Why do I carry my own microphones when I could probably get away without spending the money? Quality and control are the obvious answers. When you use whatever mic provided by the venue, the sound could be pretty bad. Eventually saxophone players get sick of sounding bad when amplified and decide it is time to buy a good microphone.
When shopping for a microphone you can get quickly overwhelmed with all the options: handheld, wireless handheld, condenser, cardiod, ribbon, tube, studio, recording, clip-on, wireless clip-on, and there are different models, many manufacturers and many are real expensive.
The place to start is Live Sound, not studio recording. Live microphones are less expensive and made to endure being banged around on stage with more durability built in. My favorite handheld mics are the classic Shure SM58 and SM57 for saxophones and all wind instruments. They are used on stages all over the world, and seem to last forever.
Handheld mics require a mic cable and mic stand to use with a saxophone. I find a stationary mic on a stand is a good option for a younger player, to work on awareness of the microphone and to learn to use mic distance for greater volume control and to avoid honking lower notes our squealing higher notes.
The only real difference between wired and wireless handheld mics is the cost in relationship to the freedom from a cable. If the musician doesn't need to move around very much, the less expensive wired handheld works great.
What about the clip-on sax mics? I love them and use them often but there are a few things you will need to be aware of. I love not having to bring a heavy mic stand, the clip-on fits in my sax case, you can move all over and the mic follows you, performance is more exciting when you can move, and they sound really good. Some band leaders ask for them and expect horn players to have good clip-on mics available.
There is a tradeoff though. The clip-on attaches to the bell of a saxophone and will also pick up the sound of keys moving as you play. First thing I do is set the EQ on the sound system to work with my sax and mic. I roll the low (bass) frequency down from straight up, to half off, or 9 O-clock. The thump from the pad hitting the sax will become very light as you roll the lows, and the saxophone does not play in the bass range so little is changed in the tone. I also roll a little bit of the highs since a small mic can sound a little bright. Mic placement always affects the tone, so you want to make sure the clip centered and low to the middle of the bell.