One of the best visual sources for the dress of working people in 18th century France is the work of Jean-Baptiste-Simeon Chardin (1699-1779). Chardin was a 'genre painter', which is to say that he specialised in depicting scenes of everyday life, involving servants and 'ordinary' (non-aristocratic people). Works such as The Washerwoman, La Pourvoyeuse, or his sketch of a servant pouring a drink give us a sense of how the working classes and lower bourgeoisie dressed. Jean-Etienne Liotard (1702-89) is another artist of the period whose work is useful in this regard - although his subjects tend to the wealthy and aristocratic, he had an incredible eye for detail in costume, so an image like The Chocolate Girl gives us an excellent insight into the dress of a servant-woman in a wealthy household.

The basics of dress didn't change much across the 18th century for any but the very wealthiest and most fashionable. For women, the underlayer consisted of a linen shift (similar in shape to a very long and baggy t-shirt), over which would be worn a pair of stays (i.e. a corset), and then on top of that a petticoat around the waist. Stockings on the feet - probably cheap lisle cotton, or maybe wool - but no knickers! This was the bare minimum in which a woman could appear, and in this she would look poor, but decent. On top of the stays, a bodice would usually be worn - not necessarily to match the skirt - and usually another outer petticoat, which we see as the skirt. Possibly also a wide jacket, for warmth. The hair was virtually always covered with a cap, in linen or cotton, maybe with lace if she had money to spend. A kerchief might be worn across the shoulders and bust, for modesty and cleanliness. Shoes might be leather, with a pair of pattens (wooden or iron overshoes) worn to protect them from the muck of the street, but the very poorest wore wooden sabots, or clogs.

Clothing for working men likewise changed little, and varied from the clothing worn by the rich only in its quality, rather than its cut. A linen shirt, a linen neckcloth, with a long sleeveless waistcoat (or 'vest') over this and a full-sleeved coat over the whole ensemble, with either knee breeches ('culottes') or long trousers worn on the legs. As with women, shoes could be leather or wooden clogs, and a hat was usually worn, often the stereotypical tricorne.

The thing to bear in mind is that 'the working people' encompasses a broad social swathe, from the very humblest members of society like street-sweepers and itinerant labourers in rags and dirty underlinen, all the way to skilled artisans and upper servants with money to spend. Servants were often especially well-dressed because it was customary for masters and mistresses to give them their cast-off clothes, which could be cut up and remade in the latest fashion. Urban workers had more access to a knowledge of fashion, thanks to print culture and to their ability to observe the wealthy.

I'm hoping u/chocolatepot can chime in here with fashion expertise! You might also find The Dress of the People by John Styles to be a useful book - although it focuses on 18th century England, much of the context is applicable to France at the same time.

As /u/OwlofDerision pointed out, it's important to bear in mind that "common people" can mean a huge range of social positions, but there actually was a certain amount of what we'd now call "traditional dress" or "folk dress" still being worn by rural people in France at this time. English-language sources frequently are biased toward treating English/Anglo-American dress (and the high-end French fashions that were imported to Britain and America) as a default, but this was just one mode. As a result, it's tough to say what the "average" person's dress looked like, since there's no average region.

For instance, in the 1661 Livre Curieux, the "woman of Paris" is depicted in a gown with a boned bodice - that is, the boning is integral to the construction of the top of the gown, rather than being worn over corsetry; the former arrangement was more common over most of the 17th century - and a skirt kilted up over a decorative petticoat, with a lace collar, shorter and wide sleeves, a muff, and a soft black hood; the woman from a village around Paris wears a bodice that laces over a stomacher, narrower sleeves, a kerchief or partlet that's tucked into her bodice and open in front, and a white cap with lappets or streamers behind; the woman from Rouen wears a black gown with a very plain, Pilgrim-like collar and a tight black hood; the woman from Cologne looks like an English woman from about fifty years before more than any of the contemporary French ones, and other northern and central European outfits depicted are similarly "unfashionable" by the high-end French or English standard, though they are fashionable by the wearer's own standard.

Notably, the young Englishwoman's illustration shows a very long point to the waist and excessive cleavage - the English were perhaps the most intent on corsetry of all the European cultures. Valerie Steele notes in The Corset: A Cultural History that travelers to England in the 18th century were intrigued by the fact that affluent women wore boned corsets pretty much all the time, and that female laborers might strip down to shift, lighter-boned or leather stays, and petticoats - in France, wealthy women might leave them off in informal dress in the country, and the peasantry might wear regional costumes that included something like the less-supportive bodice as described on the village woman above. Fully-boned stays, in France, were a style for the wealthy, the bourgeoisie, and poor urban women who saw them and were able to copy them.

Another book, Costumes Civils Actuels, dates to 1788 and shows/discusses regional dress in greater detail (also with a lot of focus on women's clothing). Here is a woman from the Pays de Caux; here is a Provençal man; here is a man and a woman from Navarre. It would be a mistake to assume that the average cauchoise or provençal dressed like this every single day - these are equivalent to fashion plates - but they show the different modes being followed in different regions. This sort of folk dress was beginning to be noticed in a positive way as it was on the cusp of being eroded: following the Industrial Revolution, urban styles were more available to those in the hinterland, and as urban styles they were more desirable, too. What followed after that can be found in my previous answer: Why are countries' traditional costumes usually from the 1800s? You might also be interested in Why doesn't England have an official national dress? and What did a traditional French maid's outfit look like?