I just bought a new house, and the drainage culvert out front along the road has several areas where the slope has collapsed, resulting in what looks like a miniature landslide landscape. What is the best way to shore this up to improve the aesthetics without causing drainage problems?
Your pictures appear to show a steeply sloped section with erosion, a flat section, and another sloped section (without erosion.) If the flat section resulted from poor compaction of an original sloped profile that did not have the flat part, you may be able to simply fill that back in so the slope is not as steep. If you want to keep it in mown grass, that might be the best bet.
Otherwise, "rip-rap" or fairly large rocks are arguably the standard method of holding a steep drainage slope.