Not sure what device to use, but I have a remote (non-ST) operated mains socket into which I can plug another main operated device - such as a desk light.
However, I need this other device to trigger an action on ST.
Preferably locally processed, in case of t’internet failure.
Not sure what to use - or how ??
Shaun
I’m not quite sure why your thread title says “sensor device” but your post discusses a desk lamp.
If you are asking if smartthings can respond to a light coming on, yes, it can. Any light that is known to the hub can be used to trigger other events in SmartThings.
As far as working locally, however, almost nothing works locally on SmartThings if the Internet is out. The one exception is the official smart lighting feature. That does have the ability to turn on light B if light A comes on, provided both lights A and B are using device type handlers which are eligible to run locally. We should also note that the SmartThings integration with the hue bridge will not work if the Internet is out. But smart bulbs which are paired directly to the smartthings hub may work locally.
Also it’s important to be aware that the SmartThings mobile app cannot talk to the hub unless the Internet is working, even if they are both on the same local network. The smartthings mobile app is a cloud-based application. It didn’t have to be written that way, but it was, so you will not have control through the mobile app if the Internet is not available to the hub.
OK, all of that said, you could use any pocket socket on your SmartThings network plugged into your existing outlet. And you could have that pocket socket turning on then trigger, say, an Osram Lightify smart bulb in a fitting on a different circuit to come on when the pocket socket came on. And that would work even if the Internet was down.
So that’s answering the question that was in your post. If you could give us more details of the specific use case we could help more on that one.
As far as a mains powered sensor, such as a motion or light sensor, that’s an entirely different question. So if that’s what you were looking for, let us know. 
Also, as far as “trigger an action in SmartThings” when the Internet is not available, again, the event options are very limited, just devices that can be controlled through the smartlights official feature. Routines do not run locally, Mode changes do not run locally, and there is no way to arm/disarm smart home monitor locally. Also no custom code runs locally. Even sunset/sunrise triggers in smartlights will not run locally. 
And for others who might come to this thread in the future because of the title, we should mention that there is an option for a mains powered Z wave sensor.
The Aeotec multisensor 6 is Z wave plus and measures light, temperature, humidity, and motion.
It can run on either battery power or with a USB connection to a mains power charger.
It should be on whatever power you intend to use at the time of initial pairing, or it won’t work correctly.
I’m not sure, to be honest, whether it is eligible to run locally or not, but most of the aeotec devices are. But you would have to use the stock device type handler, no custom code.
The older version of this device can also run on mains power and definitely does run locally
Perhaps I over simplified my requirement - I know everyone on here is quite techie, so here are the details.
Big ST setup, devices, sensors, lights, locks everywhere - works great.
I’ve installed Evohome heating (nice) - but using IFTTT to set it home/away depending on ST presence sensors - that works great - if they ever sort out integration I’ll use that instead, though not fussed.
I’ve also just installed the Evhohome Security system - its a wireless alarm and has a smart plug feature - turning off/on a wemo type mains socket according to when the system is armed/disarmed. There is no API available - or Honeywell have told me.
I now have two fobs - one for the front door lock (z-wave devolo connected to ST), one for the alarm - a Honeywell one.
What I want is one fob (honeywell) - so click this to arm the alarm, the smart socket is turned on and that sends a signal to ST and triggers the normal ST routine to lock the front door, turn off the lights etc…
So whatever I plug into that socket is the device or sensor or whatever thats hooked up to ST.
Ideally all working on local processing - but ignore that requirement if its not easily done.