Van Gogh
Picasso
Pollock
De Kooning
Joan Mitchell
Joan Snyder
Paula Monderson-Becker
Elizabeth Murray
Susan Rothenberg
Kandinsky
Georgia O Keefe
Matisse

Salvador Dali
Mark Rothko
Wassily Kandinsky
Pablo Picasso
Vincent van Gogh
Willem de Kooning
Georgia O’Keefe
Frida Khalo
Jackson Pollock
Robert Rauschenberg
Victor Vasarely
Jasper Johns
Joan Miró
Fernando Botero
I know this more than 10 but.. I love them all, they were incredible! Some of them still are.

Of course there is no definitive answer, as there are two terms in your question which are too vague to define. Firstly, as has been mentioned, how is ‘greatness’ defined? Sales? Current Value? Output? Aesthetic considerations? I’m guessing you are asking people to list their favourites or the ones they consider most famous.

More problematic is the term ‘modern’. Modern art can be said to be anything post-renaissance, which includes Rembrandt, Vermeer and David, or (more usually) the work of the ‘post- classical’ movement, which begins with Corot and the Impressionists and ends…today. Some even limit it to those artists who came after the impressionists, particularly those, such as Matisse, who eschewed realism, or figurative painting, in favour of Abstraction.

Personally, I follow the definition of the MOMA and include the art which began as a reaction to the idealised work of the classicists, which was the kind of work accepted by the Paris Salon (Paris was certainly the centre of the art world at the time). However, I have to add that British art was moving towards impressionism long before the impressionists arrived, and to leave a master such as Turner out of the gamut of ‘Modern Art’ is simply being obtuse.

So, I begin with JMW Turner, and move to the Realists, from whom we shall select Corot, who along with names like Courbet, Boudin and the Barbizon School who were the biggest influence on the Impressionists, then move to the impressionists themselves. Manet was considered by the Impressionists themselves to be their leader, though his work could never be called impressionistic. He must, however make the list if for no other reason than his importance in the development of art over the next few decades.

The impressionists themselves formed the greatest and most cohesive art movement in centuries. The question of who were the greatest of these is a subjective one and names like Sisley, Pissarro, Morisot, Caillebotte, Fantin-Latour (these last two are personal favourites) and Cassat (The American Impressionist) must be considered, as well as ’sort-of’ impressionists such as Gaugin, Seurat, and Toulouse-Lautrec. But as we must select a few we shall choose Manet, Monet, without whom there would be no ‘Impressionism’ (it was his painting; “Impression, Sunrise”, which caused the critic Louis LeRoy to name the artists ‘impressionists’ - meant as an insult), Renoir (my favourite), and DeGas (pron: der-GAR, not DAY-gas), and a special mention must go to Frederic Bazille whose work was brilliant and would be as well known as any of the others if he didn’t foolishly go off to war and get himself killed. I would also add the American John Singer-Sargent, if space allowed, simply because I enjoy his work more than any of the others listed thus far.

Of the Post-impressionists, there are too many great names to include them all, however my list shall include twelve names in case the first two are rejected as too early. So in must go Cezanne, (though I’m not a huge fan myself, his influence cannot be denied), Mattisse, (exactly the same category), and Van Gogh, (of whose work I AM a huge fan, though I’ve sold fifty times more paintings myself during my lifetime).

Picasso begins the next movement, and led the way for many years (though one cannot appreciate his work until one sees it in real life). I’m afraid I cannot include Dali despite loving his work, as neither his originality, nor his influence were as great as many think, though he would make it if only post-impressionist artists are to be included, but as the asker mentions Monet this can hardly be the case. The only real movement of any consequence since the post impressionists was Abstract-Expressionism, and my list will therefore conclude with Jackson Pollock, and the spectacular Mark Rothko.

So we have:

1/JMW Turner
2/Camille Corot
———————
1/Claude Monet
2Edouard Manet
3/Pierre-Auguste Renoir
4/Edgar Degas
5/Vincent Van Gogh
6/Paul Cezanne
7/Henri Mattisse
8/Pablo Picasso
9/Jackson Pollock
10/Mark Rothko

This is based upon influence, originality, fame, prices, and other elements. Were I to provide such a list based strictly on my own taste it would include:

Claude Monet
John Singer-Sargent
Edward Hopper
Henri Fantin-Latour
Vinnie Van Gogh
Amadeo Modigliani
Mark Rothko
Gustav Klimt
Brett Whitely (Australian)
Raymond Harris-Ching (My favourite living painter and main influence)

All of which leads me to the conclusion that the whole idea of summing up 150 years of art in a few paragraphs or ten names is patently absurd! Imagine not mentioning Kandinsky, Joan Miro, Warhol, O’Keefe, Magritte, Chagall, Duchamp, Klee, Mondrian, and Me!!!

I suggest going to
where they have a most popular artist of all time list of 30 which is updated each month. At least it’s democratic (even if I’m not in it!). According to that (and taking only ‘Modern’ artists the list would be )

Pablo Picasso 100 (votes)
Vincent van Gogh 72
Claude Monet 52
Andy Warhol 43
Salvador Dali 43
Henri Matisse 40
Georgia O’Keeffe 35
Wassily Kandinsky 30
Paul Cezanne 27
Gustav Klimt 25

go and vote for your favourite.

Pablo Picasso
Henri Matisse
Andy Warhol
Jackson Pollock
Jean Michel Basquiat
Helen Frankenthaler
Gerhard Richter
Kiki Smith
Eduoard Manet
David Wojnarowicz