Frequently, when exercisers think about core, the first thing they think about is some kind of a crunch, and the second thing they think about is a twist. That's not wrong.
Lately, however, I've been training my clients in some exercises that are just the opposite - how to use their core to keep the spine from moving. This is good for things like stand-up paddling, where too much twist will end you up in the water. It's also good for my client who has a spirited dog, to prevent injury when 60 pound Kate veers this way and that.
Here is a series of exercises with an uneven load on one side of the body, so moving that load without twisting the spine is a fun challenge for the core.
Uneven chest press
Low anchor, don't twist spine. The farther the hands get away from the resistance band, the more challenging the exercise for the core.
(not pictured - if you turn the exerciser around and have them pull downward in a paddling motion, this is a great functional exercise for stand-up paddling. I just forgot to film that segment)
One-sided chest fly - no support behind spine
This looks a lot easier than it is. If you put your back against the machine, it's all chest. Once you scoot your back off the pad, the core has to stabilize the spine before the chest can even begin to adduct the arm. To make it harder, feet closer together. Two views
"W" row with one-armed release
You need a CrossCore or other suspension trainer that has a pulley at the top. Where this becomes a core exercise is when one arm releases, yet the hips, spine and shoulders stay facing forward. Two views