Modern jack-of-all-trades cheap OPAMPs?
"All of this to say: which are the modern jack-of-all-trades cheap opamps that can work in a lot of different applications?"
I would say: there is none ! Although all opamps are in fact just opamps, you noticed that there are a lot of them, there must be a reason for that. The reason is that they all have their own purpose and are designed with a certain range of applications in mind.

You say you want a "jack-of-all-trades" but later you mention: "a signal conditioning circuit. I have to work in a bandwidth similar to audio (0-40kHz), and I need low noise and low offset (around 1 mV), and also low power."
So a "jack-of-all-trades" will not do !
Low noise audio: I'm thinking NE5532

Low noise AND low power usually contradict so you'll have to compromise, what is acceptable ? Cheap also ? Hmm, that makes it even more difficult.
You will have to search for yourself. You can play with your requirements on Digikey or Farnell, sort on price and see what you get.