How can I find out the supported webcam resolutions?

if you have video 4 linux try this

v4l2-ctl --list-formats-ext

Source : How can I list the available video modes for a USB webcam in Linux?

Two possible approaches:

Use any software which can interact with the webcam (eg, cheese), save an image and look at the resolution.

Examine the output of lsusb in a terminal, to find a line describing a webcam:

$ lsusb
Bus 001 Device 002: ID 5986:0241 Acer, Inc BisonCam, NB Pro
...

Then use the Bus and Device numbers to get more information on that device:

$ lsusb -s 001:002 -v | egrep "Width|Height"
    wWidth    640
    wHeight   480
    wWidth    1280
    wHeight   1024
...

Which should print the height, width pairs the camera is capable of - in this case, 1280x1024 plus some smaller ones.

Also possible with ffmpeg:

ffmpeg -f video4linux2 -list_formats all -i /dev/video0

Example output:

...
[video4linux2,v4l2 @ 0x7fa3a8000b40] Raw       :     yuyv422 :           YUYV 4:2:2 : 640x480 320x240 800x600 1024x600 1024x768 1280x800 1280x1024
[video4linux2,v4l2 @ 0x7fa3a8000b40] Compressed:       mjpeg :          Motion-JPEG : 640x480 320x240 800x600 1024x600 1024x768 1280x800 1280x1024
...