You can be sure to find a metro or RER train every day between 5:30am and 0.30am, wherever you might be along the line. The majority of busroutes operate from 7am to 8.30pm, some lines continuing until 0.30am. As regards to taxis, you’ll find them 24 hours a day.
The Parisian transport network has its rush hours: at the time when most residents of Paris and Ile-de-France set off for work, between 8am and 10am, and when they head home again, between 5pm and 8pm. Saturday is also a very busy day, as many Parisians use it for their shopping expeditions. It becomes increasingly difficult to find a free taxi on Saturday evenings, particularly after midnight, as Paris is out on the town!