Shallow dismissal. Singapore discriminates against LGBTQ couples as a matter of policy. Just because it doesn't affect you doesn't mean it's an anecdote.
https://the-singapore-lgbt-encyclopaedia.wikia.org/wiki/Publ...
In any case, it doesn't change anything, if Singapore is oppressing LGBT people they will do so government owned housing or not.
It's an anecdote for your argument. If the argument was "did Singapore discriminate against people through housing" it wouldn't be. If the argument is "should we have public housing [here]" then it's an anecdote. You'd have to show that this would happen in the place in question. If you want to make a general statement, you'd have to make that clear and make the case that in such countries landlords wouldn't discriminate against you for being LGBT (which they absolutely do).
Your definition of ‘anecdote’ seems way off to me. We’re talking about an objectively verifiable data point. Just because it doesn’t fully confirm some argument that someone’s making doesn’t make it an anecdote. Check the dictionary definition.
The definition of 'anecdote' has nothing to do with arguments. 'Anecdote' isn't a back formation from 'anecdotal argument'.
There's also the Fair Housing Act exemptions, where small private landlords with fewer than about 4 units, or landlords that share buildings with their tenants, can ignore the discrimination limits imposed by the FHA.
These laws could easily be repealed. The root cause of the problem is that Singapore is still a pretty homophobic society. (Also, did Singapore have same sex marriage before British Colonial rule?)
A democratic state is, ideally, controlled by the people who live under its rule. As such, services provided by the state are, to some extent, under the control of these people, who can use their political power to influence those services directly in various ways. In contrast, people don't generally have any right to change the way a private entity chooses to do business, as part of the basic definitions of property rights. Basically, people often want public ownership, and use the idea of state ownership as the mechanism to achieve that.
This difference though changes completely when discussing an authoritarian state like Singapore (and even more so in dictatorial states). The benefit of democratic control that is expected from state ownership in democratic states doesn't exist if there is no democratic control of the state in the first place (this is one of the reasons it is absurd to talk about the USSR as "socialist", when all economic activity was in fact controlled by a state that was in turn controlled by a ruthless dictator). In a non-democratic state, state ownership is just as much private ownership as corporate ownership.
Of course, the problems get even more complex when you have a quasi-democratic state that is nevertheless actively discriminating against certain groups of people.