Is using the word "singular" to describe someone or something unique an outdated adjective?

OP's example singular piece of good fortune (or luck) is a "stock phrase" that accounts for a significant proportion of all instances of singular piece, which is clearly a declining usage...

I don't think the "literal" meaning of singular as unique applies very often at all. It normally means just odd, unusual, unexpected. Usage has declined, so OP is right to suspect it's "dated".

On the other hand, singularity, which also had its heyday in the early 1800s, is coming back up again in recent decades. Doubtless partly because of mathematical singularities such as black holes, and my personal favourite, The Technological Singularity. Notwithstanding that Wikipedia link, I define this as the day when technological advance becomes so rapid that literally, yesterday's "magic" is today's "routine science", every 24 hours.

Yes, it is, although I sometimes use it myself in the comfort of my own home. It is perhaps particularly associated with Jane Austen, as in this, for example, from 'Northanger Abbey':

. . . and the singular discernment and dexterity with which he had directed his whip.

Of course, the noun, singularity, now does service in an astrophysical context.