On Firefox: Right click on the image and select "View Image Info". A new window that looks like this will show up.
Make sure your signature image is displayed in the Media Preview. Size is above that, next to the label that says size.
On Internet Explorer (or whatever they are calling it these days): Almost the same as Firefox, except instead of selecting "View Image Info" you want "Properties". I don't have that browser installed, so if anyone wants to provide a screenshot I can edit it into this post. It should be pretty straightforward.
On Chrome: This one is not straight forward. I strongly recommend either using one of the other browsers, or simply downloading the image and checking it that way. If you absolutely must use your browser, though, here's the convoluted process. Right click on the image and select inspect. It will open up the inspection element, which displays all the coded information for the entire page. At the top of the page you have a list of options, Elements, Console, Sources, etc. Click the double arrows >> to get more options, and then select "Resources". Open up the Frames folder, and the sub-folder within it, which should be the name of the page. Then open up the Images sub-folder, and find your signature image. Note: This will contain all of the images on the entire page, so you may need to scroll through a few if the URL to your image is just gibberish letters and numbers, which it probably is. When you find it, you'll see some basic information about the image, including size.