The biggest desire in the play is Blanche's. She desires more than anything else to be accepted, find a place in the world, be loved, be cherished, and be cared for. Yet she is destined never to find it, as she's much too delicate and psychologically fragile, and that's before she even arrives to stay with Stanley and Stella. Subsequent events simply make matters worse, much worse. Her desire is so strong—desperate might be a better word—that she constantly finds herself involved with men who aren't right for her, for one reason or another. It looks for a brief moment as if Mitch might finally be the man for her, but once he finds out about her sordid past, he treats her as badly as all other men in her life ever have.
Stanley desires to be lord and master of his own home, to dominate and control all those around him. That's why he perceives Blanche as such a threat; she comes into his life and disrupts its natural rhythms. Now it's Blanche who's the center of attention, not him, and Stanley hates her for it.
Stella's in the middle of all this and provides the emotional center of the play. She loves her sister and yet has no desire to return to their common past. She's made a new life for herself with Stanley, and whatever the wisdom of that decision, she's going to stick with it. Although she's regularly on the receiving end of physical abuse from Stanley, she does still have a strong desire for him, both emotionally and sexually. And now that she's about to give birth to his child, her bond with him looks set to become even stronger.