Like all things in life, it depends. If your spare tire is a full-size tire, then you can drive indefinitely, with the caveat that the tire isn’t 5-years older or more. In any other case, preferably it’s advised to drive as little on them as possible, whatever the fix may be. A space-saver spare or donut spare can be driven for a little while at low speeds, around 100 miles maximum, while a tire repaired with a kit, or a run-flat tire should be driven even less. The speeds should also be low, a maximum of 50 mph. Whatever the fix may be, always try and have the switched wheel or repaired wheel be on the back axle and not on the front axle.
Full Size Spare Tire
A full-size spare is rare nowadays. You only find them in older cars, cars with small wheels, or trucks, for the most part anyway. A full-sized spare is exactly what it sounds like: a spare that uses the exact same tire as your regular wheels. If you have a puncture, you can just switch the punctured wheel with the spare one and drive away to your heart’s desire. This means that usually anyway, you have a set of 5-wheels instead of the regular 4. Some people keep the full-sized spare with an old and beat-up tire just in case. If that’s how you plan on using it, then again, driving on it the least number of miles is advised. An old tire will lose all its traction properties due to age. Your contact patch is made out of rubber, and that rubber gets old. A 5-year-old spare is already way too old to be driven safely on long distances or periods of time. If you, however, plan on doing it properly, you need to figure out a plan on how to integrate that spare into the running set of tires so each tire can wear evenly.
Run Flat Spare Tire
A run-flat tire is, basically, a tire which has really tough sidewalls which are solid enough to be driven on. If the tire deflates, the sidewall still has enough rigidity to hold your car and provide it enough grip to at least drive it around in some fashion. The distance is a recommended 50 miles or so, maximum. The shorter the better. Driving for a while on flat tires will damage your suspension and it won’t be comfortable in the slightest. A big part of your suspension setup and the car’s overall level of comfort is the tires. You could have the fanciest car possible, if your tires are bad, your car will feel like a horse drawn wagon. Not only that, but being such an important part of the whole suspension setup, a flat tire is a missing component which the engineers haven’t factored in and shouldn’t factor in. Knowing this, running on flat tires will wear out your suspension briskly. It’s in your best behavior to limit the amount of time driven on run-flats to a minimum, but you should be safe for 50 miles driven at low speeds, around 50 mph.
Space Saver Spare Tire
A space saver tire or donut tire is just a regular old tire with the same diameter as your wheels, just a lot narrower to save space in the trunk. If it’s narrower, then the contact patch is narrower, so poor grip follows suit. This tire can be used for quite a bit longer, because it is inflated and acts like a regular tire, so driving on it for 100 miles isn’t unheard of. The maximum speed is again, around 50 mph, but it varies wildly from spare to spare. Some donut spares can withstand higher speeds while others cannot. Seeing how it’s so narrow, do not put it on the front axle. Your car’s ability to steer is paramount, and a narrower tire on one side destroys the steering feel and the car’s balance. If you have a puncture in the front, bring a tire from the rear in front and use the donut spare in the back. If you see any signs of bad weather, you should slow down even further and even stop if you don’t feel confident.
A repair kit is a small toolbox filled with rubber patches and some tools to stick those patches to the punctured tire. Those patches are used to cover whatever puncture you were unfortunate enough to get. This tire repair kit is a bit of an oddball because it needs to be quite tough all while being easy to mount. These two don’t particularly go hand in hand, so your mileage varies. You can drive safely for 20 miles or for 100 miles. It all depends on the kit and how well you can repair the tire all by yourself. It also depends on where the puncture is located. If it’s on the sidewall, you can expect the tire to last a bit longer when compared to a puncture located on the contact patch. A sidewall puncture won’t rub against the road tens of thousands of times, so it will do a better job and last longer. Driving speeds vary, but slower should be better and safer. It depends on the kit and how good of a job you think you did. The max speed is 50 mph; however, I’d advise on going even slower and driving very gently to keep the repair in place.
In a pinch, you can even use a different sized wheel if those are available due to some reason. A wheel with a diameter larger or smaller by 1 inch should work, with the caveat that they can cover the breaks and not bump into them. The same rules apply as those donut spares: drive slowly and use it on the rear axle. The car will feel significantly weirder, so you should drive even slower and do it for short distances. Also, please bear in mind that by doing so you will put an extra strain on your rear differential. Do not drive on a different sized wheel for more than a few tens of miles.