What is the current routine of modern astronomy? With about all visible stars catalogued, measured and photographed it seems that chapter is closed. I realize currently while amateur astronomy is big on cataloguing all asteroids in the Solar System, the "Big Boys" are now primarily busy with discovery of extrasolar planets. But that's certainly not all. Sure there's a lot of unique research, new, unexpected discoveries, assorted "odd" research like SETI, and so on, but I'd like to know what is the "daily bread" of every astronomer nowadays?

What are the current typical, routine tasks of an average professional astronomer? What is being observed/measured/searched for? What challenges occupy most effort and time? Or am I wrong assuming there even is such a thing, and every observatory is a unique snowflake?
What you say is not quite true: the search for exoplanets is clearly intensive, but it is far from the only things astronomers are looking at. Most of the time, in two words, the situation is: resolution &amp; wavelength. Whatever the field (if you are interested in galaxies, interstellar medium, stars and so on) you want more resolution, to resolve smaller scales (most stars are still points even with our best telescopes, or we are still far from resolving individual stars in galaxies!), to have more informations, to understand better the underlying physics. You want more wavelength, because spectroscopy gives you much more phsyical information than a single wavelength observation, for example. And to combine both is sometimes challenging: to have high resolution observations in infrared is not that easy, and it can be crucial for some fields (if you ever want to see a star forming, you better observe it in infrared, since this baby is embedded in its gas cloud that shields very efficiently its radiation).

That being said, the routine tasks of an astronomer would be


extract information from the current data. It involves a lot of coding, with Python, IDL, or more specific astronomy-oriented languages as IRAF or MIDAS. Data reduction is an important part of the job, because it is in general challenging to extract data from the raw signal you will get.
write papers about these data and the inferred informations
read a lot of papers to stay tune with the latest discoveries of other teams
write proposals to ask for more observation time/better observations/bigger telescopes
drink a lot of coffee


The three first points probably takes an almost equal amount of time for any astronomer; point 4 takes even more time for older astronomers; point 5 is also crucial for all the good things that come out of discussions over a good ol' bowl of coffeine.

Complements:

To answer to your comment and to give you an overview of the current research, I can think of:


Hershel data in infrared. People try to understand better the interstellar medium and the star formation processes in our galaxy, the formation of early galaxies, and the chemical composition and evolution of the Universe with these data.
Planck data in longer wavelength. These data are useful to understand the first age of the Universe (searching for anistropy in the CMB), but also to have an other view of the galaxy and the interstellar medium in these wavelengths.
Very Large Telescope data. There are plenty of different kind of data out of these telescope, mostly in the visible and infrared ranges, and mostly in spectroscopy. Almost everything is studied with these data, from galaxies evolution to stars in the nearby galaxies.
ALMA data in millimeter/submillimeter ranges. The same kind of objects are studied with ALMA and Herschel: early galaxies, interstellar medium and molecular clouds. How galaxies form and evolve? How stars form? In which environment? What are the dominent processes in star formation?
HESS data, in the gamma ray range. Gamma rays offer a window on the non-thermal Universe, i.e. all the extreme events occuring in the Universe. It can give precious informations about gamma-ray bursts, supernovæ, AGN (the active galactic nuclei) etc.


That's for the big projects (with a strong European bias, sorry folks I know better what's done on this side of the ocean). You can add to that all the missions to study exoplanets (like Kepler), missions to study our solar system planets (Cassini, Huygens, Messenger, Juno, all the Mars missions etc.), plus all the other facilities around the world to study anything and everything, from stellar dynamic to planet's composition. The main problem is always to understand how such structure (from the large scale structures in the Universe to small scale structures in the galaxy), object (from galaxies to satellites), phenomenon occurs, forms, appears. To understand what are the dominant physical processes at play.

Astronomy is still craving for data; the more data you have, the better your statistic will be, and hopefully the better your understanding will also be.