How many palindromic sequences in human genome?
The human genome contains approximately 1.25*10^7 palindromes longer than 6 bp (that is 8 bp or longer). Some interesting facts are also known about their distribution. 


  We found that 24 palindrome-abundant intervals are mostly located on G-bands, which condense early, replicate late, and are relatively A+T rich. In general, palindromes are overrepresented in introns but underrepresented in exons. Upstream region has enriched palindrome distribution, where palindromes can serve as transcription factor binding sites. We created a Human DNA Palindrome Database (HPALDB) which is accessible at http://vhp.ntu.edu.sg/hpaldb . It contains 12,556,994 entries covering all palindromes in the human genome longer than 6 bp. 


And no, the number of shorter ones (shorter than 8 bp) isn't mentioned in the paper because they are considered unimportant, biologically.