  We investigate several versions of the telescope conjecture on localized
categories of spectra, and implications between them. Generalizing the "finite
localization" construction, we show that on such categories, localizing away
from a set of strongly dualizable objects is smashing. We classify all smashing
localizations on the harmonic category, HFp-local category and I-local
category, where I is the Brown-Comenetz dual of the sphere spectrum; all are
localizations away from strongly dualizable objects, although these categories
have no nonzero compact objects. The Bousfield lattices of the harmonic,
E(n)-local, K(n)-local, HFp-local and I-local categories are described, along
with some lattice maps between them. One consequence is that in none of these
categories is there a nonzero object that squares to zero. Another is that the
HFp-local category has localizing subcategories that are not Bousfield classes.
