Church is really the classical place. It's a shame there's not a similar place for large groups of secular people to gather once a week for social events.
I'm a Catholic so I attend Church often and in my experience, it really depends on how the Parish runs; in some places, people literally show up 5-10 minutes before Mass and then rush to the parking lot as soon as it ends, so you never have a chance to speak to anyone.
However, in contrast, my fiance is from Peru and goes to a Latin Church; there you have a very strong sense of community and can meet plenty of people. Also, you probably won't be meeting your SO at the Church directly but you will most likely be introduced _through_ someone from the community (especially older people who have bigger families).
The other thing about religious communities is that "dating" is not about hookups or casual sex but more about about finding life long partners and marriage. This can be a blessing or a curse depending on what stage of life you are in (though if you are truly devout, you won't mind) so it's not perfect.
Anyway, for non religious people, there are still plenty of ways to meet: Volunteering programs, social clubs, sport centers, etc.
The only advice I would give is meet as many people as possible because chances are your potential partner will be introduced to you through friends of friends eventually.
Unitarian Universalist Church. Ethical Culture Society. There are quite a few non-credal churches or "not-a-church" organizations built along churchy lines.
I would argue in the current climate "political activism" and "political groups" have replaced God and the Church as a religion. Secular, certainly, religious absolutely. I've heard from some people that they won't even consider dating someone who has even an iota difference in political views. If this doesn't sound like religious dogma I'm not sure what would.
Not sure if it exists in your country, but all over Europe there are "Civic Centers" that host various cultural events, activities and curses where you can meet people from all walks of life. Great place to meet people with the same interests as you and usually hosts relatively cheap events as it's meant to be attend-able for all people.
Are we talking about paying for seats to spectate? Cause if so - I highly disagree. You might get lucky every so often but I found people generally keep to themselves/their friend group.
