They are a lot of them. Here you can find some sources:

• Structure et exploitation de la conjugaison des verbes en français contemporain – Claire Blanche-Benveniste (2002)
• Une description bilingue des temps verbaux :
  étude contrastive en corpus – Cristina Grisot & Bruno Cartoni (2012)
• Pistes pour l’enseignement de la conjugaison – Suzanne-G. Chartrand (2011)
• Les temps du passé français et leur enseignement – Emmanuelle Labeau and Pierre Larrivée (2002)

I did not read all of them but they seem relevant.

To give you a perspective of a native French, as a child, we are taught verbs in 3 groups:

• Group 1 includes all verbs in -ER such as ‘manger’, ‘chanter’, etc.
• Group 2 includes -IR verbs such as ‘finir’ with the present participle ending in -issant (ending) or with present for we ending in -issons. We know, from hearing other people talk, which verbs are in this group. ‘Courir’ is not in this group because we can not say: ‘nous courrissons’.
• The 3rd group includes all the other verbs like ‘prendre”, “voir”, “faire”,… as well as -IR verbs whose present participle does not end in -issant. For this group with a beautiful book for that called: Bescherelle that you can look through when you don’t know the conjugation.

We start to learn conjugation: the indicatif (présent, futur, passé composé, imparfait, …), then conditionnel, then subjonctif.

Why in this order? Indicatif is the most common. Conditionnel implies a notion of probability (it could happen). Subjonctif is for more complex sentences with relatives.

Some tenses are commonly used:

• présent
• passé composé
• futur
• imparfait (Tu faisais quoi ?)
• présent du subjonctif (Faut que tu prennes ton permis)

The others are mostly used in literature. You can hear some plus-que-parfait in formal French. You will never hear futur antérieur, passé antérieur and passé simple in informal French and rarely in formal French (almost never).

The futur antérieur and plus-que-parfait are less used but nevertheless mainstream: Tu avais pris quoi ? / Tu aurais pris quoi ? (Thanks to jlliagre in the comments).

Here is something really interesting: most of the people do not use all of the tenses but can clearly understand why we use them and understand the timeline. Those tenses are used a lot in literature and give important details to the action. Passé simple means the action is short, whereas the imparfait means a long action. Plus-que-parfait could mean that the action was before the past action, and so on. They are really important. It’s just that in everyday French, we just use a few of them.

For example, this sentence:

“J’étais parti (plus-que-parfait) quand elle m’appela sur mon téléphone (passé-simple).”

In informal/common French, people could say:

“Je suis parti (passé-composé) et après elle m’a appelé au téléphone (passé-composé).”

Instead of leveraging the plus-que-parfait, we can use après to tell that the second action happened after the first one. Obviously the second one is less pretty but both are understandable.

In the end, you can say most of the things with présent, futur and passé composé but for the beauty of the language, you can go deeper.

For the impératif, you know them because of your parents: Range ta chambre., Lave-toi les mains, …