I have some sucker branches and they look kind of OK in my book, I like nature.
BUT
I've recently learned that suckers branches reduce the amount of nutrients that go to the main parts.
Attaching two photos from my garden.
The definition of a sucker is a shoot or branch growing from below the graft; the graft point may not be visible above the soil, as in roses for instance, so it's best to check by pushing soil away to see whether the shoot is coming from above or below the graft. If it is from below the graft, then it should be removed, preferably by wrenching off if possible, because it means it's growing off the rootstock and is not part of the grafted plant above the graft. This does not apply to un-grafted plants, such as your Cordyline palm.
In the case of your Cordyline, growth at the base, whether it's off the main trunk or from below ground, is not uncommon and sometimes indicates a problem higher up the plant, perhaps on the main trunks, or that the plant has been stressed and has produced new growth at the base in order to survive. I'm not seeing any obvious problems with the upper growth on your Cordyline, but you might want to inspect before deciding what to do about the basal growths. If there's no problems, you can either leave the basal growths to grow on, or locate the base of each and just cut them off. If some have any root material of their own, these can be grown on separately after detaching them as long as they have some root material left intact.
In the case of your Hazel tree, check the point of origin for the straight stem nearest to the fence, but the others are not suckers, they are water shoots or just new stems because they arise off existing wood above ground. They can be removed by cutting them off close to the trunk using sharp, clean loppers or pruners, and would be best removed.
If you're fine, they are fine. Suckers (coming from the roots) or water shoots (coming from above ground, near the base) do need to be removed from grafted plants since they come from the rootstock and will take over from whatever has been grafted on. Apart from this, it is purely a matter of taste. Water shoots are part of the natural growth pattern of plants like hazel. They get removed because people have a fixed idea of what a tree should look like, but if you want to appreciate the natural form of the plant there is no need to do anything.