There is alot of fear about tubeless tyres but in reality they are the easiest tyres to live with. As not all tyres are created equal. tubeless tyre choice is important too but the majority of tubeless issues road users have are user error simple as that. Below is a guide to avoid user error. I have worked this all out by trail and error (alot of error on my part as well).
First of tubeless tyres should be run with sealant. Some insist on riding without sealant. Riding without sealant can be done on proper UST rims and with a few tubeless tyres, but I would not do it, too much flint in Suffolk to take that risk. There are many brands of sealant, Orange Seal, Stan's no tubes, Effetto mariposo, Zefal, DT Swiss, Schwable even IRC do one. they all work in similar way. The sealant is alkaline and this keeps the latex in solution. Mixed in is a particulate like gliter (orange seal use that I think), something like cornmeal is used in stans (it may not be cornmeal but it has a similar size and texture). When you puncture the sealant is forced into the hole and comes into contact with the air. this cause the pH to drop and the latex falls out of solution. The particulates in the sealant then stick to the latex plugging the hole. This of course means if you use CO2 to inflate your tyre most of the latex will fall out of solution. Stans for example will turn in a white watery liquid when CO2 is used which is about as useful as chocolate teapot, in fact less useful, I can eat a chocolate teapot. If you use CO2 to inflate tyre just mounted add more sealant through the valve core, the tyre should remain locked to the rim so you can then inflate with a track pump.
Of course because CO2 causes the change in pH can also get a tyre that has a puncture that wont seal to seal. I have used this trick before then added more sealant when I get back home.
Sealant dries out slowly. It needs to be topped up. I use effetto Mariposo sealent now mainly because with CO2 it does not turn into a watery liquid, the latex will fall out of solution but no water is left meaning when fresh sealant is added it is not diluted. I top up my IRC tyres every 3 months or so. 30ml in a road tyre is sufficent. I drain sealant though the valve core through an open syringe with gravity. I dont bother checking how much sealant is in the tyre normally I just pour some in.
If you keep the sealant topped up you will find it will seal most punctures (if not all) pretty quickly. You can get a fair bit of airloss but even if you drop to 30 psi you can still ride on that. so you can stop and faff with a small pump or continue riding and use a track pump at home. I have done the latter as I dont often have a pump with me I come to that later. This is the problem with 23mm tubeless tyres. The air volume is small and therefore the pressure drop is large before the tyre seals. Bigger tyres can seal at a higher pressure due to the larger air volume. This make larger tubeless tyres more practical. I consider 25mm the minimum size for a tubeless tyre for this reason alone.
Sometimes however the sealant just cant seal the hole. Most of the time that is because there is insuffient sealant in the tyre but sometimes it is because the hole is too big. At tis point many people get out that spare tube and faff trying to get the tyre off, fit a tube while getting sealant every. then swear alot and ruin their thumbs trying to get the tyre back on again. No wonder some give up on tubeless tyres after that. There is a solution though. In my back pocket is Loctite flexible superglue and normally Genuine Innovations tyre worms. The worms are the string of butyl rubber and some with an applicator. For a bigger hole in the main body of the tyre I first try the superglue. If that can fill the hole and allow the tyre to hold air great if it leaks I try get the worms out. You place the worm with the middle in the applicator and shove it into the hole. This will plug it but a worm by itself is not always enough to seal at high pressures. Now get the flexible superglue out again. Smear that over the worm and hole. Let it set then inflate. I would be wary about inflating to high pressure straight away. I would put enough air in to get me home and no more. When you get home you will want to trim the worm and you can then make sure there is enough glue covering the hole before trying higher pressures. Remember a 25mm tubeless tyre is run at not more than 90 psi anyway.
Tubeless tyres can be repaired if they have a butyl or latex lining. IRC's tyres do not many other do though. If you have a proper lining then you use a inner tube patch and vulcanising rubber glue to seal the inside of the tyre. For tyres without a lining you may need your superglue again to get the patch to stick. You can do this if your plug/superglue fix starts leaking air.
Sidewall cuts can be dealt with using the flexible superglue. I would not try a tyre worm as they make the hole bigger. not something you want to do with a sidewall cut. Small cuts/holes can be filled that way. you may not be able to hold high pressures but you can get the tyre to hold enough air to get you home. One sidewall cut I got when riding my IRC's off road on a damp dark night (asking for trouble) was fixed this way. Initially it could only hold air for a couple of hours at 60 psi but I kept on apply the glue and after a week or two it could hold 80 psi indefinatley and the tyre did another 3000km without issue. As it was a 28mm tyre 80psi was more than enough as I run those at 70 psi.
So what you dont do is fit a tube to fix a puncture issue with tubeless tyres. If you carry a tube ask your self why, the hint is in the name TUBELESS!!!!!
What I carry;
When I am commuting on the commuter I have tyre worms, flexible superglue, a pump, CO2, valve core remover and a 2oz bottle of sealant.
When riding my race bike I just carry flexible superglue and CO2 unless I am feeling paranoid.
The thing is even if I carry what I do on the commuter bike it still takes up less space than tubes and levers. I do not carry a tube or tyre levers. As far as I am concerned my tubeless tyres once fitted will remain on the wheel until they are worn out, unless I have to patch it which thankfully I have had to do only once.
So I hope that helps answer questions about riding with tubeless tyres. I have given up on clinchers with tubes and won't be switching back.