I would like to search for cheap flights for a week-end out, without specific restrictions on dates or destinations. Is there a flight search engine that allows one to search for all flights leaving from city X to an unspecified/flexible location on an (also unspecified/flexible) Friday evening and returning on the next Sunday?
Sure this can be done: have a look to azair which has very powerful search options (see the advanced options), including choice of return and departure day, length of stay, period within which the trip should happen and a range of departure and arrival airports including "take me anywhere."
A similarly powerful search option for the anywhere part is kiwi.com.
Note: much of this is copied from my own answer here, plus you might want to take note of this overview of flight search engines I once composed.
With Skyscanner you can search for flights from a city to any particular destination. You could manually search through every upcoming weekend like this.
I'm not aware of a flight search engine that allows you to show a list of these flight searches all at once.