There is some cycling infrastructure in Madrid. It was a bit difficult to find. Usually, it consists of bike paths on shoulders of roads separated from pedestrian traffic by distance, trees, and then sometimes by street furniture.
On some streets, the bike lanes went along the centre median. Unfortunately, the intersections were poorly handled with cyclists needing to go back to the sidewalks for crossing streets.
Time wise, it was very inefficient resulting in many cyclists staying on the road. The first discovery of suitable streets for cycling, was a booklet handed out by the tourism offices. It listed 10 routes for cycling. Others, I found by cycling across them, especially the new path system on roads in the suburb of Leganes — approximately 15 kilometres south of Madrid. Bike signals were commonly found throughout the city where presence of cyclists may occur.
Bus lanes were marked for taxi and moped use. Never found out if cyclists should also be using these facilities. I did observe some cyclists staying in the bus lanes and remaining in the adjacent general traffic lanes. Certainly, the bus drivers on Calle del General Ricardos, did not seem to mind me being in the bus lane. I think one driver was trying to tell me to get off the lane or something, but then his Spanish was well beyond my “si” and “adiose” level.