To me there are five true classics-
Margaret, Mary, Catherine, Ann, and Elizabeth.
As far as classy, I think of adding Sarah, Rachel, Rebecca, Caroline for quite timeless names.
Some that fall in and out of favor and fall victim to overuse in certain decades but send a classy vibe are:
Vivian, Veronica, Victoria
Alexandra, Anastasia, Angelica, Allison, Anna
Bridget, Bella, Beatrice
Calla, Charlotte, Cora
Dahlia
Eloise, Ella, Eliza, Eva, Emma
Francesca, Frances
Genevieve, Grace
Halle, Holly
India, Ingrid, Isabel
Jessica, Joanna, Johanna, Josephine
Kate
Leah, Laura
Molly, Miranda
Nadia, Natalie, Nora
Olivia, Olive
Priscilla, Portia
Rose, Rosemary
Stella
Tabitha
Zara, Zinnia