My older daughter and wife saw bears on more than one morning run in Sequoia National Park. I don't recall if they saw any in Yosemite. They saw baby bears on or near the paths and the mother no far off. My family would just stop, turn around, and go back the other direction. We typically ran in the parks first thing in the morning when there were few vehicles in the parking lots. I assume the bears are use to encountering people on the trails in the national parks. They saw bears on more than one morning. I never saw one. We spent 4 to 6 days in each park.
This spot is notorious for bears. Some get a little too used to tourists.
Northern Minnesota. Out for a jog down a shady road at a Scout Camp, bear saw me before I saw it. It spun around and dove into the brush. Man, they move really fast. I kept running towards the spot not breaking stride, but had to change my pants later.
If you let out a holler or a whoop every minute or so, you can avoid surprising a bear. They'll know something's coming down the trail.
I have come across a few koala bears while running. Fortunately I haven't yet encountered a drop bear.
Surprisingly, for living a lot of my life (and running since 1978) in the "north woods" populated with black bears, only one encounter while running... and that was two summers ago (August 2015).
About 75 yd ahead of me on an off-road RV trail, a smallish black bear rumbled across the trail. Think were both startled at each other's presence.
And during a family camping trip 40+ years ago, I managed to stay asleep in our tent while a bear was in our campsite looking for scraps, with our dog barking and my dad yelling and banging things together to scare it off.
3 times, all black bears. First was in the Sierra Nevadas, an adult and 2-3 cubs playing on a lakeshore 20m away. They ignored us.
Second and third times were both solo bears in central PA.
One crossed the trail nonchalantly about 50m in front of my group, ignoring us the whole time. Large bear.
The other time I was running solo on some trails and came across a bear right on the side of the trail, less than 10m away from me. It scampered away from me up a hillside.
No point trying to run away from bears as many of the posters suggest they did, as you'll never outrun them if they want to catch you.