It’s technically legal, hazardous, and deeply unpleasant. I would personally not ride there. Neither would almost anybody else, judging by the
Strava heatmap.
MCBC is working on improvements, but it sounds like those are decades out, if they ever arrive.
But I’ve also been
underwhelmed with other routes between Solano and Marin/Sonoma counties. The stretch between Napa and Petaluma isn't manifestly unsafe, you're just riding in the shoulder of a windy, trafficky, same-y major road. And while I haven't ridden from Vallejo through American Canyon myself, I hear tell that it really sucks. (If anyone has info to the contrary, I'd be very eager to hear it.)
What's the bigger picture of what you're trying to do? Many years ago I learned the hard way that if the answer to your question is "Tequila!", you probably ought to ask a different question. Maybe if the answer to your question is "CA-37!", the upstream question similarly warrants revisiting...
- Drew