Republic generally is seen as the best (any production bonus from Communism does indeed come too late to counteract it), and if you're playing very warlike where shields due to no war weariness + military police are more important than research, Monarchy can be the best; again, Communism would come too late to be worth the switch. If you're playing warlike enough for Monarchy to really be worth it over Republic, you'll probably have conquered a good chunk of the planet by the time Communism comes around as-is... especially because you'll probably prioritize Replaceable Parts for Infantry + Artillery, and maybe even go for Tanks and Bombers, before going for Communism. Some players may argue that Republic is always better than Monarchy, but my take is that depends on playstyle and difficulty level. The lack of war weariness is certainly a point in Monarchy's favor from the annoyance factor, and it's not an unusual government in warlike games.
Communism suffers greatly from the fact that while it balances out corruption, most large empires aren't equally developed when the switch into Communism. So while the weak outer cities become more productive, they still aren't well-developed and are still fairly poor; meanwhile the rich inner cities all of a sudden have far more corruption than they did, and if you aren't prepared ahead of time (buying at least courthouses with gold as Republic/Monarchy in outer cities, ideally police stations, factories, market, etc. to boost shields and commerce), the decrease from inner cities can be worse than the benefit from outer cities, resulting in an actual decrease in GDP and production upon switching. I ran into this in my PTW Russia --> USSR game; my science rank tanked from 60% to 20% as maintenance stayed the same but income plummeted. I wish I still had that save; it was one of my most interesting Civ3 games because of that element (and the Brits almost winning a space race).
Feudalism is very niche IMO. Due to the 2nd Rule of Governments, it's almost always a good idea to have switched to either Monarchy or Republic first, which would mean two periods of anarchy, violating the 1st Rule of Governments. So this leaves two situations:
1. Got to Middle Ages without researching Republic or Monarchy. Generally considered a bad idea. The 3-food/shield/commerce penalty of Despotism is very severe, so sticking with it until the Middle Ages is almost never a good idea; if so you've violated the 2nd Rule of Governments.
2. Are in Monarchy and Republic and switched. Switching to Feudalism at this point violates the 1st Rule of Governments. And if you're in Monarchy, you're probably being warlike, in which case the war weariness of Feudalism makes it a bad idea. If you're peaceful, however, the commerce bonus of Republic almost certainly outweights the unit support of Feudalism.
That said, there are rare cases where it can make sense. If you're being crushed by unit support in Republic, and need those units for military police/wars, it can be worth switching. Generally if you have that many units and are at war, Monarchy is worth considering instead due to the lack of war exhaustion, but if you really have gobs of units, Feudalism does have more support (although you should probably send those units to the front!), and if the war is temporary, the war exhaustion can be acceptable (but be sure you really can't afford them, and need them, before switching). In other words, the niche cases tend to be ones where economics dictates Feudalism, and there simply isn't an alternative to keeping all those units.
I used Feudalism in a game in 2010, and from what I can tell/recall, it was only because for some reason I didn't wind up researching Monarchy/Republic. I didn't have enough units to justify it (although I had plenty of cities to generate support), and bogged down in tech in the 2nd half of the Middle Ages. Eventually I switched to Republic (in the late Middle Ages) for the commerce.
So what's that leave? Democracy, for one, which as mentioned almost never provides enough benefits to be worth it. It's arguably better in Vanilla than Conquests; in Conquests Republic has double unit support costs, but has enough free units that by the time Democracy is available, a Republic usually has less than 1 gold per unit in actual military support costs, or will soon once Sanitation hits, and thus Democracy is more expensive. In Vanilla, both Republic and Democracy have no free units, but also don't have double support costs. So while Democracy is equally good in Vanilla, Republic is typically better in Conquests late-game than Vanilla late-game, since its unit support will generally be higher in Vanilla. Even so, it's hardly ever worth switching to Democracy in Vanilla.
Last and least is Fascism. It gives higher unit support than Communism, but Communism already has plenty generous unit support; you'd almost have to try to max out Communism's unit support. And its penalties - minus one pop per city upon switching, and no culture growth until you have majority culture in a captured city - are very high. Supposing you were switching in the Industrial Era due to too high of war exhaustion in Republic/Democracy/Feudalism, it's a tough case to make all around. Communism has lower corruption in the long haul, and no penalties, although in the short haul Fascism's corruption model may be better. But if communal corruption doesn't work for you, then why not go Monarchy and avoid the Fascism penalties? By late game, Monarchy's unit support is usually enough anyway.
And I suppose Despotism deserves a mention. In rare cases, the 4/4/4 unit support can make it worth staying in Despotism until you've planted a couple more cities to help bump up what Monarchy/Republic would support. It's rare, and when in doubt switch, but I have had cases where despite getting rid of the Despotism penalty, I've tanked my economy due to too high of unit support. Typically this is when I have a larger army than I usually do early - such as when I'm launching an early warrior/archer war on a neighbor that's kind of far away, and have a bunch of units marching over there. However, once either a few more cities are planted, or the enemy's cities are taken, Monarchy or Republic will be better, so staying in Despotism is only ever the best in the short term, not the long term. The other case, of course, is where you're a couple turns away from finishing a wonder that someone else might finish first if you switch to Anarchy - but in that case make sure to switch as soon as it's finished. Yes, I have been that player who stayed in Despotism to finish a wonder, and then forgot about switching governments after the wonder was completed
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As for Anarchy... you'd have to have an extremely unusual setup for it to make sense. It's one saving grace is there's no unit support or building maintenance. So, it could make sense in a case where you build up a stockpile of gold, then built up a humongous army that cost way more per turn than you could afford, using the stockpile to keep you afloat, and then started a war with the huge army but couldn't afford its upkeep. In that case, Anarchy would allow you to keep the army in the field, although without/mostly without reinforcements. It is still possible to produce things in Anarchy (Civil Engineers/Citizens/Scientists/Tax Collectors), but it's painful and very inefficient. I can't think of a case where practically it would make sense to build up such a huge army that Anarchy would be the only way to support it before starting a war, but theoretically it could make sense if you could find a scenario justifying that big of an upfront buildup. One of the biggest buildups I've done was 50 galleys + 100 Medieval Infantry/Pikemen, and even then Monarchy made more sense. But it's fun to think of a scenario where Anarchy might be the best option, even if only for a short time.
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. (Sorry for the lateral off-topic but just couldn't resist. Cheers.)
