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.
For all n up to 50000 I have computed terms of the aliquot sequence starting with n until it terminated, entered an aliquot cycle, or reached an entry with at least 80 decimal digits. The aim now is to continue until at least 90 digits (or termination) is reached. All this is ongoing work as part of a bigger effort by several people (see the aliquot page of Wolfgang Creyaufmüller, or the page of Juan Varona, or the aliquot page of Paul Zimmermann).
Here is a summary of the status of my own computations.
For sequences starting beyond 50000 check the aliquot page of Paul Zimmermann.