Nice examples are worked out in Audin ([2004](https://ams.org/mathscinet-getitem?mr=2091310), §VI.3.d), Arvanitoyeorgos ([1999](https://ams.org/mathscinet-getitem?mr=1708918))([pdf](https://www.emis.de/proceedings/7ICDGA/IV/)), McDuff-Salamon ([1998](https://ams.org/mathscinet-getitem?mr=1698616), §5.6). It’s not true that $\omega$ is rarely explicit: e.g. on all coadjoint orbits (including $\smash{S^2}$) it is, and DH gives a formula of Harish-Chandra for their Fourier transforms; see Berline-Vergne ([1983](https://ams.org/mathscinet-getitem?mr=733806)), Vergne ([1983](https://ams.org/mathscinet-getitem?mr=713793)), or Guillemin-Sternberg ([1984](https://ams.org/mathscinet-getitem?mr=770935), ends of §§33 and 34).