There are seven mother sauces in "classic" French cuisine. Careme had four, one of which didn't overlap with Escoffier's five. At some point after Escoffier touched Adam's finger and gave humanity life, everyone slapped themselves on the forehead and said, "merde." Merde is not a mother sauce, but mayonnaise is.
Careme:
Allemande
Bechamel
Espagnole
Veloute
Escoffier:
Bechamel
Espagnole
Hollandaise
Tomate
Veloute
Modern, Synthesized:
Allemande:
It's pretty much an egg stiffened veloute. No one uses it. It started petering out of French cuisine by the mid 19th C around the beginning of the culinary revolution and trend towards simplicity which carried Escoffier to sainthood. It enjoyed a brief resurgence right after WWI, with a bunch of dishes (especially fish) which everyone called Parisienne then, but nobody does anymore. Now Parisienne seems to mean anything but.
Bechamel:
Darn near everyone uses bechamel. The issue is, uses it for what. Bechamel, Espagnole, and veloute, as roux based sauces have pretty much disappeared from the high end haute French, some other high-end European, and New International Cuisines, but they're still going very strong in a lot of regional and bourgeois cuisines.
Espagnole:
It's been rendered pretty much redundant, especially as a path to demi. During the nouvelle and California revolutions we discovered that if you left stock on the stove it thickened up by itself, and that was pretty much it. I like Espagnole as a mother, but haven't been a professional cook for multiple decades; so I don't count. A few people on CT use it -- one of whom employs it not as a mother but as gravy. Titomike is using my version to make demi, and is very happy with it.
Hollandaise:
Mmmm. Hollandaise.
Mayonnaise:
Where would the Japanese be without it.
Tomate:
The mother tomato sauce, "tomate," wasn't something you tossed on spaghetti, or used right out of the pot. Rather it was used to supply structure without starch as well as some color and sweetness -- the sweetness coming after it married the other ingredients and cooked down. Everyone still uses a tomato "sauce" for the same purpose, but no one uses Escoffier's version or anything like it because canned tomato products are so good there's just no need. Modernly, we use tomato paste and go from there.
Veloute:
It's pretty much gone from modern high end French cuisine, New International Cuisine, and so on; but is very much alive in ordinary cooking worldwide. Think of it as gravy and you get the idea.
Vinaigrette*:
NOT a mother sauce, for two reasons. Vinaigrettes should be made a minute, because they eventually separate, and the eventuality doesn't take very long. Daughter vinaigrettes are still vinaigrettes, there isn't enough distinction or progression in the daughters. Mostly though the daughter is created at the same time as the mother, which drives a stake into the heart of the whole mother/daugher relationship as I understand it.
Distinguish all that from, say, taking jarred mayonnaise, thinning it with diluted vinegar, and sweetening it with sugar in order to make "Alabama White Barbecue Sauce."
But Bluesed got it from Lynne's who got it from someone else -- which means at least there's some following for the idea that it's a grande. Plus, I'm not researching this, just pulling it out of my bony head; and I certainly don't know everything. There's something on which we can all agree.
Hope this illuminated for someone,
BDL
PS. It seems our OP, who pleaded help ASAP, has forgotten us. Que lastima.
PPS. I published a slightly edited version of this, called Snow White and the Seven* Mother Sauces, to my blog.