The main open question in the area is given by the Catalan-Dickson conjecture, which states that for any starting value n one of two things happens: either the sequence ends in a cycle, or it reaches 1 after finitely many steps (and is said to terminate then).
Cycles of length 1 come from perfect numbers: 6 = 1 + 2 + 3 = s(6), 28 = 1 + 2 + 4 + 7 + 14 = s(28), etc. Cycles of length 2 are also called amicable numbers: 220, s(220)=284, s(284)=220. Longer cycles are also called sociable numbers.
Progress in numerical work to verify the Catalan-Dickson conjecture relies upon progress in integer factorization methods, as no better method is known for the determination of the divisors of a number than the obvious method using its prime factorization.
I became interested in aliquot sequences while testing the integer factorization algorithms in Magma. I have listed some of the numerical work I have done. Several other people are involved in a big effort to push our knowledge for small starting values.