What exactly are we seeing here? Are these little spots around the edge of the membrane or in the middle? Or is the big dark rectangle the issue?

Either way, something is not going well. I agree with Subhajit Karmakar about the blocking. If the small dots persist too, make sure everyone who uses the membrane wears gloves and uses clean scissors for cutting.

Echoing others, what am I looking at? I see some sporadic dots on a membrane and one that appears to be completely black. Is this correct? Echoing the first two, the black membrane hasn't been adequately blocked, if at all. You'll want to use a solution of 5% milk in PBS with 0.1% Tween-20. Block AT LEAST 40 minutes. Longer for whole cell lysates and you'll want to just spike your primary into the blocking too. My personal experience is that this does help cut down on nonspecific binding.

As for the membrane that looks like it doesn't have any dots/bands (I think there is a membrane there), this could be caused by multiple things:

1.) Your primary antibody may be bad/old. Not all clones work for WBs. Check the manufacturer's website and make sure the one you're using is validated for WB use. Although the standard dilution for primary is 1:1000, you may need to do a titration to find if you need a higher or lower dilution to get a clean looking WB image.

2.) You did not incubate the membrane in primary long enough. Overnight at 4 C is what I've always done, although sometimes you can get away with RT for 2 hours.

3.) You used the wrong secondary antibody. If for example, if the host for your primary is mouse, make sure the 2nd antibody is anti-mouse.

4.) You may have reacted all the luminol (if you imaged for a while) and there is no more chemiluminescence for the imager to collect.

I hope this helps you! Good luck!