I have a couple of questions,

In the first picture is the runoff coming from your neighbors yard? It looks like the slope runs from left to right and then toward the bottom of the picture
In the second picture it looks like all of the water from the front of the house on that side runs down this slope. Is that correct?
Ditto for the third picture.
Do you have gutters and downspouts?

Do you have gutters and down spouts,

Yes, we have had gutters and downspouts since the house was built. The downspouts dump all water into buried drainage tubing. A lot of the rain runs from the front yard, down the side yard slopes to the backyard. It does not collect in the backyard. It washes the side yard grass away every season. I see know way to stop it.

The root cause of the erosion is not just the volume of water that flows down the slope, but the speed of the water. Most any groundcover, be it native plants/shrubs or turf, is going to struggle to gain hold with the speed of the water. It is a good thing the gutters are channeled elsewhere or it would have been worse.

Another factor that exacerbates the problem is the ground around the house appears compacted (is it clay?) and does not readily absorb the runoff.

There are several options open to you, but you may need the services of a landscaper who can evaluate the situation on-site.

1.      Terrace the slope – This would work well for the area in the third picture. You could make each terrace into a little rain garden.

2.      Install swales (depressions) across the slopes to catch the water and channel it to the side

3.      Install a French Ditch meandering across the slopes like a stream to catch the water and channel it down the slope to the bottom of the hill.

4.      Install rain gardens on the slope to catch the water and reduce both the flow and the water velocity.

Any plantings that you may do choose native plants and grasses. Natives have deep root structures that help absorb water this lessening the runoff.

If none of the above are attractive, you could seed or sod, but you need to do some soil prep. If the soil is compacted you will need to work in some organic material to break up the soil so it will absorb water more readily. You can then sod or seed. If you sod, lay the sod across the hill and stake it until the roots grab hold. If you want to seed, use the seed mats. Again lay them across the slope and stake them down.

But as I said earlier you need to solve the root cause of the problem and that is the volume and speed of the runoff. Hope this gives you some ideas.

If you have any other questions feel free to ask.

Thanks Paul