The number of metastable vacua is not known exactly, but commonly quoted estimates are of the order 10500. See M. Douglas, "The statistics of string / M theory vacua", JHEP0305, 46 (2003). arXiv:hep-th/0303194; S. Ashok and M. Douglas, "Counting flux vacua", JHEP0401, 060 (2004).
L. Susskind, "The anthropic landscape of string theory", arXiv:hep-th/0302219.
S. M. Carroll, "Is our universe natural?" (2005) arXiv:hep-th/0512148 reviews a number of proposals in preprints dated 2004/5.
M. Tegmark, A. Aguirre, M. Rees and F. Wilczek, "Dimensionless constants, cosmology and other dark matters", arXiv:astro-ph/0511774. F. Wilczek, "Enlightenment, knowledge, ignorance, temptation", arXiv:hep-ph/0512187. See also the discussion at [1].
M. R. Douglas, "Statistical analysis of the supersymmetry breaking scale", arXiv:hep-th/0405279.
V. Agrawal, S. M. Barr, J. F. Donoghue and
D. Seckel, "Anthropic considerations in multiple domain theories and the scale of electroweak symmetry breaking",
Phys. Rev. Lett.80, 1822 (1998).arXiv:hep-ph/9801253
H. Baer, V. Barger, H. Serce and K. Sinha, "Higgs and superparticle mass predictions from the landscape", JHEP03, 002 (2018), arXiv:1712.01399 .
M. Tegmark, A. Aguirre, M. Rees and F. Wilczek, "Dimensionless constants, cosmology and other dark matters", arXiv:astro-ph/0511774. F. Wilczek, "Enlightenment, knowledge, ignorance, temptation", arXiv:hep-ph/0512187. See also the discussion at [1].
Motl's blog criticized the anthropic principle and Woit's blog frequently attacks the anthropic string landscape.