Not much, regional languages are almost not spoken anymore. Until the second half of the 20th century, the government actually fought against regional languages, it was prohibited to use them in schools for example, children were punished and shamed for speaking their native language (it is known as "la vergonha" in Occitan, a regional language that was spoken in all southern France). Nowadays, there are almost no native speakers left. There are bilingual schools in Breton, Catalan, Basque and Occitan, but teachers aren't native either, and even though they know the language very well, they have a thick French accent, which explains why modern Breton/Occitan is heavily influenced by French (if you listen to recordings of elderly people whose first language is actually Breton/Occitan, you will notice that the pronunciation changed radically, due to the lack of native speakers). Nowadays, those languages are mostly a cultural heritage, and aren't really spoken outside clubs and bilingual schools. You might still find a few elderly people in very rural areas who speak those languages, but really, it is very very rare. To give you an idea, Occitan was the mother tongue of almost 40% of France in 1860. Nowadays, approximately 5% of French people claim to have "some knowledge" of the language, and among them, there aren't probably a lot of people who actually speak it, let alone who speak it natively.
So to sum up, those languages almost went extinct, were reintroduced kind of artificially, and are now alive in very little circles, and most speakers aren't natives.
Indeed, France doesn't officially recognise any other language than French: "Art. 2 : La langue de la République est le français." - Constitution
The region where Occitan is the most healthy is in Val d'Aran, in Catalonia, Spain. Aranese (one of the dialects of Occitan) has an official status there, and is still spoken natively by a part of the population.
The kind of writing we see in your picture were pretty common, some people who went to school in Brittany during the early 20th century reported that there were signs saying things like "Il est interdit de cracher par terre et de parler breton" ("It is forbidden to spit on the ground and to speak Breton").
some special bilingual schools are created for example "diwan" schools where students learn Breton, TV shows, radio programs are shown in medias (for example in "corse " language) , linguistic policies are being established but it's a complex question because local people and French government often disagree about it