While I was weeding today, I noticed in several instances small daylily plants anywhere from six inches to foot from any other main clump. I can't tell where the little plants belong to. I know I dropped some seeds last year.
I also know they have large root systems. This is the second year for about half, and the others the first year. (Planted last summer and fall) One plant in particular I bought as a single fan and planted last fall has a tiny plant about eight inches from it. I am not sure if it belongs to that daylily or not. What are other people's experiences?

Some daylilies do have "runners" and they are not often welcome in the garden because they tend to spread into the neighbor daylily clumps. If you dropped some seeds that of course could be where the new daylily fans are from. There are also a few daylilies that have a tendency to multiply pretty fast. That is normally considered a good thing.
I think as long as the little new growth daylilies are not the result of "runners" the new plants will be a welcome addition.

If you planted Fulva, a species Daylily, it propagates on runners that can appear a long distance from the parent plant. Fulva doesn't have seeds though. Most Daylily fans will show up pretty close to the parent plant. I haven't experienced anything other than Fulva and Kwanso sending runners. That's why I dig them every year but they always come back with more.

I don't have any ditch lilies or wild type in my yard. All of mine are cultivars, about 95% are registered. I have a very small handful of unregistered plants.
I just thought it was weird. Hopefully they get big enough to figure out which clump the tiny fans belong to.

IrisLily,
Here is an older thread that might shed some light on the stray plants. There are still registered plants that have that trate of sending out rhizomes "runners".
The thread "Rhizomatous daylilies?" in Daylilies forum
Here are a couple of photos I took a few years ago, I had a few cultivars that still processed this trait.

I think this is actually a rhizome, the mulch I think was so thick it was acting as soil. So this "runner" would have been a rhizome I think even though it was on top of the soil, but under the mulch. I used to call them stolons, but I was told stolons run on top of the soil and rhizomes run below the soil level.

Those actually look like like a normal distance away from the clumps. I have many Daylilies that propagate like that. When the fans get bigger, and throw a scape, you'll know which group they belong to.

The bigger clumps I can sort of understand. The single fan in the first picture I just planted last fall had me a bit puzzled. It is actually closer to the iris than the "mother" fan.

@Seedfork

Larry, That looks a lot like a stolon to me and not much like what daylily rhizomes look like.
You wrote that it formed under the mulch. The mulch was on top of the soil. What developed was under the mulch but on top of the soil. I assume that the mulch surrounded the daylily clump and actually touched the daylily clump above the soil. I expect daylily rhizomes to grow a length and then have a slight constriction every now then. I also expect them to be able to produce roots and shoots along their lengths, particularly where some of those constrictions are located.

I would call yours a stolon. I would say that it probably developed from a bud that was under the mulch and that sprouted to produce a new "increase" fan. Finding itself in partial darkness (because of the mulch) but not as dark as being completely underground it elongated to produce a long extension of itself (a crown - which is a stem). So what it produced may not have been rhizome-like but more stolon-like. Normally, those extension growths would start to form underground in the soil and in the dark and be a rhizome. The extension growth on your daylily may be similar to the extension growths of the crowns that developed after I buried entire fans under mounds of soil (The thread "Planting daylilies "too" deeply" in Daylilies forum)

I would still call Larry's picture a rhizome - it was below the surface and has nodes (see pic). If daylilies typically produced stolons then we would see them running along the soil surface more often when there is no mulch, I don't recall seeing that ever. Crown, rhizome and stolon are all stems, the distinction between the latter two is whether they run along the top of the ground (stolon) or below (rhizome). If the fan at the end of Larry's rhizome was lopped off and the rest planted it may well grow roots and shoots at the nodes. It does look rather narrow at this point but doesn't look to be very old as it hasn't suberized yet so is still whitish.

If the plants are quite tiny the simplest way to tell what they are is to dig them up and see if they still have part of the seed attached. Usually it is very close to the surface of the soil. I get lots of seedling popping up if I don't collect all my seeds or some fall. I usually dig them up as I find them and pot them. But if I don't see a seed I replant them where I found them because they are probably just a fan that popped up away from the main clump.