I don't know the exact answer, but what I would try is, open the Recording tab of the Sound properties. Make sure the Realtek audio driver is installed because the audio jacks work better with it. If the Realtek driver is installed you will see the words "Realtek High Definition Audio". If it is not installed you will see "High Definition Audio Device".

After the Realtek audio driver is installed, right click on a blank area of the Recording tab and thern select "Show disabled devices" and "show disconnected devices". Look for external mic under those. If there, enable it and set as default.

If the external mic is not there, then go back to the Recording tab and set "Microphone" as the default recording device then look under "Properties" and "Configure" and look there for an external mic setting. Also make sure the mic volume is turned up.

I tried what you suggested, but I seem to lack line in or similar device, only microphone and some kind of mix one:

In microphone, under both configure and properties I could find no specific setting for external microphone. Could a driver update fix this? Can I diagnose it on some other way?

I seem to lack line in

Look for it under both "show disconnected and show disabled devices.

Oh, by the way, the mic jack might not work if you have the laptop setup for 5.1 audio like in the picture here.  Try opening the Realtek Audio Manager and check that, then click on the Microphone tab and see what kind of options are there.

only microphone and some kind of mix one:

Stereo Mix is a recording source that lets you record and capture sounds playing through the laptop, such as streaming audio off the internet. It has other uses too.

Could a driver update fix this?

Maybe. Try the HD audio driver right off the Realtek site instead of the Dell provided one.

Can I diagnose it on some other way?

I don't know how you have been testing it so far. The normal way is to open Windows Sound Record and speak into the mic and see whether or not your voice gets recorded. Also, Microsoft has a mic test.

There are also mic tests in Dell Diagnostics audio tests.

Also a mic test in Dell's other tests, but I don't know if they test the external or only the internal mic.

Another thought. When you first get a laptop and open the Sound properties, the default is to see Microphone ande Line In, but to not see Stereo Mix. You do not see that so I suppose someone has made changes to your Sound configuration. There is nothing wrong with that, but if you want to get back to the defaults just do a simple re-installation of the audio driver. Go into the Device Manager, right click and select uninstall, do not select the option to delete the files, and re-boot. Windows will re-build the driver from the files on the hard drive and restore its default configuration.

Did you make sure that it is NOT set up for 5.1 audio as I mentioned above? When set up for surround sound it will output audio through the mic jack.

Other than that, download and install a fresh copy of the audio driver. If that does not fix it then it is a hardware problem -- the impedance sensing circuitry.

Same machine, same problem.

The "jackstick" part of the Dell Audio recognizes that I plug in the microphone jackstick, but it shows up as "speakers", and indeed, I can hear sound-output if I connect headphones.

My guess is that this is a software/driver-issue, but hard to figure out... hoping for a resolution.

Yes, it's definitely set to stereo. I already installed a fresh copy of the dell-drivers. I have also tried with multiple microphones. I'm still hopeful it's just crappy software not recognizing the mic.

(slightly different laptop though: XPS L521X )

I have the exact same problem with my XPS L521X, the hybrid mic/headphone jack does not work with an external mic even testing multiple microphones that work on other computers just fine, it is recognised as headphones. After talking to Dell's Tech support, I've used Dell's factory restore, done a fresh OSRI twice once to Win 7 Ult then to Win 7 Home Prem just incase it was a compatibility problem with Win 7 Ult, Each time I reinstalled ALL drivers before testing and I've tried both the latest audio driver from Dell as well as Realtek. Dell replaced the motherboard then even sent me out a USB key to do a Dell restore of the OS again and still no joy with the mic jack. I've searched everywhere on the Internet and there's not much info about this problem out there. I have another date with Dell's Tech support today to see if they can figure it out how to get it working again.

Looks like this thread has shifted from being about the XPS L502x model to the XPS L521x. What applies to one does not necessarily apply to the other. The L502x has a regular headphone jack socket plus another one that doubles as a s/pdif port, plus a regular mic jack.

The L521x has one regular headphone jack plus one combo jack, and no regular mic jack. The combo jack is a smart phone type combo jack. You can use a smart phone headset (combines both headphones and mic into one plug) or conventional headphones in the jack. You can't directly use a regular computer mic because the contacts won't mate. To use a regular mic you need an adapter like the Headset Buddy below. Notice the 4 segmented plug. One segment is the ground. Two of them are the headphones. The tip is the mono mic (no stereo line-in with a combo jack). Somehow it does not surprise me that tech support does not know this stuff.