Question

The solar neutrino problem has been "solved" by discovering that neutrinos have mass and they oscillate. So how accurate are now our predictions about the number and types of solar neutrinos that reach the earth?

Answer 1

The solar data analyzed with current solar models are in good agreement with reactor and atmospheric results for and . Nor is this just the opinion of one small-brained experimenter: at least some theorists concur.1

This is not, however, a very strong test of the models.

References:

The big SNO paper. SNO probed the solar neutrino spectrum in detail. On page 34 you'll find their spiffy comparison plot of flux for electron and muon flavor channels between their results; the "standard solor model" with no oscillation; and Super Kamiokande. They are several sigma from the un-oscillated model, and agree with SuperK.
The most recent KamLAND oscillation paper. The plot you want here is figure 2, which compares the combined SNO-and-SuperK results to the KamLAND results in- space. Note that the allowed regions overlap. Also on the arXiv as 0801.4589.

So, we conclude that SNO sees neutrinos fluxes consistent with an oscillated solar model where the oscillation parameters are derived from atmospheric and reactor data.

However, the error bars on the oscillation parameters are non-trivial, so it limits how strong a test of the solar model we have.

I presume the SNO paper can be found on the arXiv as well, but I don't know its number.

1 The paper looks like the opening talk given at a solar neutrino workshop or perhaps at the appropriate session in a broader conference.