Is there any way to stop a rose from suckering a lot? I have a 'Moonshadow' and it spends way more energy suckering than putting out the legitimate rose. I've got a few others that sucker a bit too.

I don't know how you handled the first suckers that appeared, but if you cut them off, you only stimulated the rootstock to put out more suckers. They have to be torn off the plant to be removed entirely. Dig down to the point where the sucker originates and pull downwards on the base of the sucker to remove it from the plant.

How do you deal with suckers that come up from the roots some distance away from the plant?

Those might not be from the rootstock. I have some gallicas that sucker and I used to have rugosas that did that. Those suckers sometimes come up a few feet away from the original plant. If they are from the rootstock of a grafted rose, they're probably from a forgotten rose that you removed, but not entirely. I've taken out some Dr. Hueys and I've been surprised a few years later when they reappear.

I cut the suckers because I have never been able to tear them off.

The suckers that are coming from the roots (no root stock) are from Seven Sisters and I really don't mind as I love the rose and have plenty of space. Some of them pop up 8' away or more.

If you have the room, Porkpal, you can afford to encourage that. I never had enough room for the rugosa suckers, and they were so thorny!

Liz, by cutting them, you're producing the same stimulating effect that any type of pruning would produce. I'd be tempted to dig up the entire rose, pull off all of the suckers, and re-plant the rose. This is a good time of year to do that.

There is a chemical called "Sucker Stopper". I've used it on trees but never on a rose, but it might be worth a shot.
It stops about 75% of the suckers on my crabapple trees.

Do you get that at the nursery or online?

I buy it at local nurseries, but here it is at Amazon:
https://www.amazon.com/dp/B007RPAAW4/

It cost several dollars less than that last time I purchased it locally.
Big box stores might carry it.

Thank you

I have a question about suckers too. I have four that came up on my Cecile Brunner climber this year. Do they stay on or do I remove them? for any help!

Climbing Cecile Brunner is rarely grafted, so these probably aren't suckers from the rootstock. Are you talking about the four tall vertical canes in your photo? Those are just new canes. That's how the plant climbs.

They just looked so different from the rest of the stems. I've had other climbers but not this kind where the stems generally bow over. I will look forward to the new growth!