Two stops in China

Yes, it definitely is since your entire transit time, counted from your scheduled arrival in Beijing, until your scheduled departure from Shanghai, is shorter than 24 hours. In other words, you are definitely eligible for 24 hour visa free transit and you do not need any kind of visa.

Regardless of what I have written, I am very worried about your very short connecting times... You have to be aware that you will have to approach, upon landing in Beijing, at 24/144 hour visa free transit counter where you will apply for a temporary entry permit because your next flight is a domestic one. This application is for free and the procedure is fast because you will present them your eligible flight itinerary. However, you will have to be extremely fast and literally run, run... When you arrive in Shanghai, you will arrive on domestic flight and you will have to cross the immigration (exit China) and appear on time at the gate for your flight to Singapore. Honestly, if I were you, I would rethink this flight option and maybe accept it only if it is on a through (single) ticket... In any case, I advise you to look for some alternative with longer connecting times.