The Pi Zero is so minimal, it can be tough to tell if its working at all. Here's how to do a quick check (from this sticky)!

• Take your Zero, with nothing in any slot or socket (yes, no SD-card is needed or wanted to do this test!).
• Take a normal micro-USB to USB-A DATA SYNC cable (not a charge-only cable! make sure its a true data sync cable!)
• Connect the USB cable to your PC, plugging the micro-USB into the Pi's USB, (not the PWR_IN).
• If the Zero is alive, your Windows PC will go ding for the presence of new hardware & you should see "BCM2708 Boot" in Device Manager.
• Or on linux, run sudo lsusb or run dmesg and look for a ID 0a5c:2763 Broadcom Corp message. If you see that, so far so good, you know the Zero's not dead.

I tested on Linux and here's my actual dmesg:
[226314.048026] usb 4-2: new full-speed USB device number 82 using uhci_hcd
[226314.213273] usb 4-2: New USB device found, idVendor=0a5c, idProduct=2763
[226314.213280] usb 4-2: New USB device strings: Mfr=1, Product=2, SerialNumber=0
[226314.213284] usb 4-2: Product: BCM2708 Boot
[226314.213288] usb 4-2: Manufacturer: Broadcom

but there will be NO LED LIGHT (looks so dead but its alive!)