Except in the case of massive social upheaval, I think co-ed showers will never be the standard. As @Samuel said jokingly in a comment:
Probably when men stop excitedly asking.
This is quite a humorous comment, but there's a much more serious way to put it - separate bathrooms and showers will always be around as long as rape is still a thing. One rape is all it can take to bring a costly lawsuit down on the heads of whatever organization owns the building.
The only place for co-ed showers is one in which everyone using them feels safe. It's hard to imagine a society in which every single woman feels comfortable having men be able to see them naked without the woman being able to do much about it. Men can also be victims in situations like this - there are male victims of rape.
Co-ed showers can be put in place in limited locations, such as the "clubs and venues which target alternative lifestyles" that @HenryTaylor mentioned. The people that go there expect it, and choose to use them because they still feel safe there.
I wouldn't be surprised if society moves in the opposite direction that you're thinking - rather than having a single co-ed shower, we'll have more single-occupancy showers. It doesn't make as much sense from an efficiency standpoint, but what is there about legal issues that would make you think efficiency is anywhere on their minds? Also think about this question - how can the mother of a 6 year-old girl tell the difference between an honest transgendered person and a committed pedophile man? The best way for a facility to protect itself from lawsuits from both the mother protecting her child and a transgendered person wanting to use the bathroom they think is the right one for them would be to have single-occupancy and family bathrooms.
Even space constraints alone won't bring about co-ed showers - if you have enough room for two showers, there's no reason you can't have a divider between them that turns it into two bathrooms. So you've either got a single-occupancy bathroom, or enough room for more than one.
A dystopian government, like the one @MrDracoSpirit mentioned, might impose co-ed showers on certain groups of undesirables, but I would not count that as becoming a standard. That government would need to keep enough people happy to prevent a complete revolt, and those people would not be forced into co-ed showers. So rather than becoming a standard, co-ed showers would be seen as a sort of punishment.
@MrDracoSpirit's third point is something that would definitely make co-ed showers likely. If culture lost interest in sex, then rape would most likely stop being a thing. And that, as I said earlier, is something that would make co-ed showers reasonable.
@Samuel Well, I can honestly say that pretty much every Mormon area of the US doesn't really approve of gays or "third gender" people, but it is less discrimination than just weird looks and a little bit of avoidance. I mean, I think people need to accept emotionally, mentally, and physically that here are literally two genders. Male and female. Deal with it. – Xandar The Zenon – 2016-04-12T04:12:46.547
@XandarTheZenon Wrong. There are literally two sexes, but gender is far less of a dichotomy. You can disagree as much as you like, but it's not your business to tell other people what gender they can or cannot identify with. Deal with that, or don't, it doesn't matter to the facts. Who cares what the Mormons think about sexuality anyway? – Samuel – 2016-04-12T04:46:01.860
@Samuel Before I continue this argument, let me just say I am a lot more forceful with my opinions and the truth than most, while also a lot more argumentative. So don't judge others based on me. I'm not telling you what you can or can't do, I couldn't care less. I'm Just stating facts. Male or Female. – Xandar The Zenon – 2016-04-12T05:03:32.680
30Probably when men stop excitedly asking. You've just set us back a bit. – Samuel – 2015-06-02T15:33:51.290
@Samuel Good point. You have my deepest apologies. – DaaaahWhoosh – 2015-06-02T15:36:53.003
5"These days, people are becoming more and more accepting of new ideas of gender and sexuality, many of which reject the male-female binary." And pretty much all of those people are exclusive to niche websites. The vast majority of the current human population will give you weird looks and/or roll their eyes if you try telling them stuff about a "gender spectrum". If co-ed bathrooms were to become a normal sight, I'd imagine it'd be for a practical reason, not a cultural one. Heck, disabled toilets are already co-ed. – Pyritie – 2015-06-02T16:18:19.217
3@Pyritie Except that it isn't. The really off the wall ones, like "I self identify as a stapler" are rather niche. But the mainstream of "I don't identify as a man, I identify as a male" are rather widespread. – corsiKa – 2015-06-02T17:13:50.847
3@corsiKa: San Francisco and tumblr do not count as "widespread". – Pyritie – 2015-06-02T17:15:25.367
5There is more to it than just San Francisco. Dozens of countries (Australia, Germany and India to name a couple) have official recognition of non-binary genders, and all support for it is within the last 15 years which indicates it's the beginning of a movement. Given Germany's influence in the EU, and France's influence as well as their open views on sexuality, it is very likely you'll see non-binary status recognized in most European countries in the next 20 years. – corsiKa – 2015-06-02T17:20:35.067
4@corsiKa is correct. This is a very common (and widely accepted) thing outside of the southern United States. – Samuel – 2015-06-02T18:04:22.210
@corsiKa that sounds like the beginnings of a pretty good answer. If you want to post it as such (maybe with some sources or more examples), I'd definitely upvote it, and maybe even accept it. – DaaaahWhoosh – 2015-06-02T18:23:05.860
3@Samuel: As someone who lives in a very progressive city in the northern United States, I disagree. It's more common, but still not common. And definitely not "widely accepted". – Andrew Coonce – 2015-06-02T18:32:46.320
1@AndrewCoonce There are still pockets of bigotry, of course. Wisconsin, while I've never heard of any part of it being described as "very progressive" (though I don't doubt it at all, look at Austin, Texas), is barely out of the bible belt. So, I'm not surprised by your observations. Do you have that joke where you say the only thing wrong with Madison is that it's surrounded by Wisconsin? – Samuel – 2015-06-02T18:48:53.867
I'm not a sauna goes myself, but AFAIK mixed gender saunas aren't uncommon in Germany. – CodesInChaos – 2015-06-02T18:58:09.843
@Samuel: I think you're looking for "Madison: 77 square miles surrounded by reality". Madison is frequently mentioned with cities like Austin and Portland, but even here there's probably ~40% chance that a given person will here "I'm a trans-whatever" and unconsciously roll their eyes. – Andrew Coonce – 2015-06-02T19:00:19.403
7I'm wondering if it's not headed the other way. It seems more and more that showers in locker rooms, etc. are becoming more and more: "doesn't matter gender - but I want my own room/stall". – Mikey – 2015-06-02T19:11:38.310
@Mikey I agree completely. For the general public this is certainly the case. I think the co-ed thing mainly appears in films depicting the military (actually Starship Troopers is the only example I can think of right now). The military has an agenda concerning individuality, so forcing everyone to look the same, sleep in the same room, and shower in the same room just serves that agenda. – Samuel – 2015-06-02T19:30:36.677
@Samuel As for appearances, I can think of Starship Troopers, the reboot of Battlestar Galactica, Robocop, and maybe Alien(s). So yeah, mostly a military theme. – DaaaahWhoosh – 2015-06-02T19:33:29.930
1@Samuel Wisconsin is so far north of the bible belt that it hasn't even seen it for the past 20 years due to the large belly it has developed in the form of Illinois, Indiana, Ohio, Kentucky, and half of West Virginia and Tennessee. Any farther north and you'll literally be in Canada. – TylerH – 2015-06-02T19:49:53.083
3This question appears to be off-topic because it is asking about predictions for social norms in the real world, not about world building as defined by the scope of the help center. – TylerH – 2015-06-02T19:53:04.750
@TylerH You may be right, though I promise it is for worldbuilding purposes. – DaaaahWhoosh – 2015-06-02T19:59:59.357
@DaaaahWhoosh In that case I would recommend editing the question so that it is asked within the context of building a world. – TylerH – 2015-06-02T20:01:21.073
It will be 6 to 8 weeks. – PyRulez – 2015-06-03T00:05:27.893