Spring and fall are both good times to divide perennials that have gone astray. Spring however, makes it a lot easier to have success in the propagating of more tender species. Many plants love to be thinned so they can be rid of a crowded root zone. How they are thinned depends mostly on the type of roots they have.
Tuberous roots, like dahlias, love to be dug up every few years for a refresh. Often times, a few tubers on the cluster will rot and get mushy which eventually will spread to the rest of the gang. With a short handled digging fork, remove the whole plant. [Take a moment and notice which way is up so you can remember how to set the tubers back in the ground. Sometimes a stem from last year is still apparent and a helpful guide]. Shake off as much of the soil as you can. Remove any rotten tubers and gently pull apart any tuber clusters that will separate from the mother plant with ease. Some single tubers may come apart, these are also fine to re-plant as long as they are not damaged, or rotten.
Perennial herbs, like Echinacea and Valerian, can also be dug up and gently divided. Using your hands or a sharp knife should do the trick. With these types of roots, you can usually see the individual little starts as you separate the root mass. Valerian will often have little rooted pups ready to go along the old stem. Set these smaller starts in a 4 inch pot with potting soil if you want to ensure good growth. Larger clumps can go right back in the ground. This propagating technique is much faster than starting from seed and super affordable!
Perennial groundcovers can have runners (a slender stolon that runs along the ground, sprouting roots from the nodes) like a strawberry plant, or simply form a rooted mass along the soil surface like some varieties of thyme. Detach and plant the rooted nodes off of runners — simply cut an appropriate sized section of your rooted mass of groundcover.
Succulents are some of the easiest plants to propagate from cuttings. Without even having to ask nicely, they generally start sprouting roots right off of their growing stems. Remove a few sets of leaves and set directly into a new home. If there are no air roots, simply lay succulent leaves on a bed of lightweight potting soil or seedling mix that is slightly damp. In a few weeks, roots will sprout and you can then plant the rooted leaf.
Most perennials will re-seed with ease. Look closely at the ground around your favorite hellebores for volunteer seedlings. They naturally want to colonize so if you prefer to keep them from taking over, when they are done blooming, cut off the seed pods.