Minimum number of trainers to win raids

Short Answer: Yes, a single trainer can win a 3-star raid but they need to have decently strong Pokemon. 5-star and mega raids typically require at least 3 very strong trainers, although certain Pokemon that are double-weak to a single type can be defeated with only 2 very strong trainers.

Long Answer: This is a difficult question to answer because it very much depends on trainer and Pokemon levels, types, and moves. 1-star raids are easy to beat once a trainer gets Pokemon above level 15. 3-star raids can be beaten by a single trainer if their Pokemon have types and moves that take advantage of a raid's weaknesses and are at least level 30. With the increase of Pokemon levels up to 50, it should become relatively easy for a single trainer to beat a 3-star raid as long as they can power up a couple Pokemon to the max level, but the stardust and candy requirement will prevent most trainers from doing that for a while.

5-star and mega raids will probably always require more than one trainer to beat, even with the level increase to 50. They just have too much health for a single trainer to take down, although I have seen "mock solo raids" where two people enter a raid and only one person attacks, so they can get the attack boost from trainer friendship levels and beat the raid. But these require max level Pokemon with weather boosts and only work against certain raids that have low defense or double weaknesses (ex: rock against a fire/flying Pokemon, fire against bug/steel Pokemon, ground against fire/steel Pokemon).

For normal trainers, 5-star and mega raids typically require at least 3 people, and the high defense raids like the Regi trio will require 4 trainers. I suggest trying for 4-5 trainers for all 5-star and mega raids to make the fights easier and more of a certain victory.

Yes 3-star raids can be soloed. In fact thus far, the only 3-star raid in Pokemon Go's history that could not be soloed was Shuckle, which was before level 50 and Mega evolutions were introduced. Using Pokemon with STAB super-effective coverage against the Pokemon, with strong move, a high Attack stat and as high a level as possible are recommended. The degree to which this needs to be applied varies based on the difficulty of the raid Pokemon. For example, Sharpedo is an easy raid boss which can be potentially beaten at trainer level 20, and Alolan Marowak requires a strong team but is still very doable. A satisfactory team is always a requirement, but weather boosts, the opposing Pokemon's moves, and luck also affect the difficulty. General tips for level 3 raids outside of the choice of Pokemon and moves can be found on the Pokemon Go Press website.

5-star and mega raids almost always require multiple people. Technically speaking, the minimum number is still one. A select few can be soloed, for example Deoxys-A and Mega Abomansow with a sunny weather boost and a very strong team including Mega Charizard Y. However, most of these raids require at least 2 or 3 people depending on the boss difficulty; even very difficult level 5 raid bosses like Regice can be trioed (but it's very tough!) This 2 and 3 people threshold usually requires strong teams and sometimes friend boosts, weather boosts, and/or luck. In practice, most people don't have strong enough teams to complete the duo or trio, so the number required is more. This number depends on the difficulty of the raid boss, your team, weather boosts, the opposing Pokemon's moves, luck, and also friend boosts. With a team of level 40 players, I agree with Kadima that generally 4-5 trainers is adequate for most 5-star and mega raids.

The simple answer is that most 3 star raids, except for Shuckle, can be completed by a single Trainer, though the recently increased Pokemon levels may overcome Shuckle. If you are having difficulty, here are a few factors to consider, in order of significance:

1. Use Pokemon with advantageous types and moves.
2. Power up your Pokemon. Although long term, consider #3 before powering up since stardust is the most precious resource.
3. Use Pokemon with better IV stats.

Answering the more general question about minimum number of Trainers for 5 star and Mega raids is much harder because the answer depends on each raid boss. Silph Road has run simulations and created a webpage, Pokemon GO Raid Bosses, with color-coded guidelines on the difficulty of each raid boss. Using the guideline is a matter of comparing and estimating your situation with the simulations, based on PokeBattler, in two ways: Pokemon and battle style. The guideline suggests a variety of possible Pokemon, but your inventory is what you will actually use. Your battle style, in terms of how much dodging/attacking, will also differ from the simulations. If you want a fine grain estimation, use PokeBattler directly to simulate your Pokemon against a raid boss.

You can get a rough estimation through empirical experience to compare against the guideline. If you complete a particular 3 star raid, then look at the color code to gauge your level of effectiveness. Completing a few different raids will give you a good average. Then you can estimate whether you can complete other 3 star raids. Extrapolating to 5 star and Mega raids, you can then estimate how many Trainers at your level are needed.