I think it's important to consider the time-length a gift. It's a chance to adjust my initial impression of you walking in the room. By that I mean when you enter the space, I'm naturally going to assess your look. This guy looks too tough for this part; that woman looks too plump to play Martha, and too old to play Honey. But then you get the chance to readjust my preconceptions of these characters.
1 minute monologues can reveal a lot about your craft. The first thing I'm going to find out: your choice of monologue. Is it relevant? Were you smart enough to choose a monologue that allows me to envision you in my play. A male auditioning for Streetcar's Stanley would be smart to do a monologue from Shanley's Danny in the Deep Blue Sea or Miller's A View from the Bridge.
Also choose a monologue with levels. The best monologues have a moment when the character discovers something that changes the tone. If an audition is a microcosm of a full performance, show me you can navigate through beats. That impresses me more than a monologue with one emotion.
Show a beginning, middle, and end. I'm certainly not the first to make this point. It's clean and sharp to show a beat before your first word. This isn't a term paper - no need for an attention getter. You're in the middle of the room, my attention's on you. Also, once you finish, give it a second to resonate in the room. That's a perfect package in my mind.
Now, Adam, you bring up cattle calls. There are success stories about actors being loud in a unique way to grab the part, but I say keep it simple. Show your absolute best during your tiny share of time. You can never guess what will catch the interest of a director in a cattle call. It's what it is: a haystack with a few needles. You're either what I'm looking for, or you're hay...or cattle...I don't know...but being the loudest won't necessarily get you picked by me.
I think 3 people have told me that they knew a large black man who sang "I Feel Pretty" and got the part. In fact, I think I read it in a book too. I think it demonstrated that he could handle the text against the challenges of type. If you're going to think outside of the box, make it good. Otherwise it's annoying and a waste of time.