If the Singularity is coming in our near future, then we should expect it's already happened multiple times around our own galaxy. If Singulatarian arguments are to be taken seriously in their strongest form, and events post-singularity are in principle unpredictable to us as we are presently constructed, then those alien singularities could look like anything, including nothing. This would explain the Great Silence, but it would be an unsatisfying explanation: "something happens to everyone that turns them into something we wouldn't understand even if we were looking right at it". It's been supposed that post-Singularity intelligences would want to deconstruct large bodies (like planets) to allow more particle interactions and increase computational abilities. Therefore we might look for strangely dusty systems, like 59 LY-away Eta Corvi, which is not by any stretch a still-forming system but has two belts of dust: a Kuiper Belt analog, and a warm dust belt within 3.5 AU that should disappear within 20 million years - and the age of this F-class star system is about 1.5 billion. More mundane mechanisms have been advanced (here in the Astrophysical Journal) to explain the persistence of the inner dust belt, which is chemically very interesting - but again, in the strongest interpretation of the Singularity, there's no point in trying to guess what any anomaly is, because we couldn't understand it.
A better alternative is to assume that we just don't know what we're listening for (yet, but that eventually we might), or that the absence of electromagnetic signals detectable at Earth doesn't mean no intelligence, or that we haven't looked in the right places, which especially applies to the search for von Neumann probes, which may seek out material agglomerations in low gravity (comets and asteroids). An artifact may be worth many signals.