UPDATE:
The strange sprouts coming from the roots of my roses have not changed since I posted on January 4th.
Yesterday, I had a "Eureka" moment. I was cleaning up under some mature Cotoneaster parneyi shrubs and discovered some of the same feathery sprouts! Then a few minutes later, I discovered more of them under a 5-year-old Camellia. The three locations are 30-40 feet from each other. The roses are in full, all-day sun. The other two locations are filtered sun (small trees nearby). All locations are well-drained soil with mulch added and are covered with layers of bark nuggets. They all get irrigated briefly every other day.
Porkpal suggested Mistletoe. I doubt that's what these are, but I'm beginning to think they are epiphytic or parasitic. If I see any flowers, I might be able to ID them.
Another weird thing is how much these sprouts look like a plant called Rhipsalis. That can't be what they are, but they sure look similar.
I'll post with any other developments.
G D
