To the best of my knowledge, an anomaly is anything that can't be explained by current theory, while a singularity is very specifically a point where space-time is so compressed that the local gravity is so high that the escape velocity exceeds the speed of light, i.e. the center of a black hole.

Yup just this.

Though in popular modern pseudo science / futurism "the singularity" has also taken on the meaning of a merging of the biological and mechanical / digital.

Really? What a gross subversion of terminology.

Bugs me to no end too. I understand that words can change, but coopting meaning is sort of a pet peeve of mine.

One is a tower defense, the other is a shooter.
...
Wait, that IS what we're talking about, right?

Okay, I actually laughed a bit there.

In mathematics, a singularity is a point where a function is undefined or poorly behaved. For example, the function
f(x) = 1/x
is undefined when x = 0; that is a singularity.

In physics, I believe that singularities happen when the mathematical models used have singularities; for example, at the center of a black hole where the mathematics breaks down badly (assuming general relativity and quantum mechanics).

Thank you Siri from Wikipedia.

Wait, you ACTUALLY LAUGHED? At one of MY jokes?!
That's a singular anomaly right there.

A singularity is defined as a point at which a function has attributes that are infinite. So a black hole has infinite density and thus is a singularity. Likewise at the universe's current rate of expansion it will spread infinitely far and become a singularity.

A technological singularity occurs when humans build an AI smarter than themselves - which in turn builds another ai smarter than itself in a shorter amount of time - which in turn does the same thing. This keeps going until technological growth becomes infinite - thus a singularity. Humans may merge with an AI to become something else - or we could get left behind as refuse. Transcending biology isn't what defines a technological singularity - but it might be necessary for continued existence at that point.

1. Singularity.
2. Anomaly.
3. Anomaly.
4. Anomaly.
5. Neither, it is just a happenstance. It would have collapsed anyway I think.

You're welcome.

EDIT: I don't get all this talk about black holes and dividing by zero. The only black hole I know is in Uranus.

I thought that was a brown hole. But I don't know, that may be a misunderstanding similar to how we Europeans commonly say "black tea" to what the Asians call "brown tea".

You're the best, like the way you know things;)

Anomaly: Something is not as usual/normal. It's anomal.

Example: Water has the highest density at 4 degrees Celsius above freezing point. That is not like other fluids behave. It's an anomaly.

Singularity: Something goes to extremes at a certain point. It's either extremely small or extremely large.

Example: Black holes have an extreme mass concentration. The logarithm goes to -infinity for argument equalling zero.

In terms of fantasy and sci-fi you often have great freedoms. You can make up the world and define what is normal and what is not normal (so anomalys can be almost anything). Singularities have to exhibit some sign of extremity though and additionally they may be anomalies.

For example a black hole could be an anomaly too if it would be an anomal black hole (a wormhole maybe) that is different from the usual black holes.

TL-DR: Anomalies are never normal and Singularities are always extreme.