We calculate very long low- and high-temperature series for the susceptibility χ of the square lattice Ising model as well as very long series for the five-particle contribution χ⁽⁵⁾ and six-particle contribution χ⁽⁶⁾. These calculations have been made possible by the use of highly optimized polynomial time modular algorithms and a total of more than 150 000 CPU hours on computer clusters. The series for χ (low- and high-temperature regimes), χ⁽⁵⁾ and χ⁽⁶⁾ are now extended to 2000 terms. In addition, for χ⁽⁵⁾, 10 000 terms of the series are calculated modulo a single prime, and have been used to find the linear ODE satisfied by χ⁽⁵⁾ modulo a prime. A diff-Padé analysis of the 2000 terms series for χ⁽⁵⁾ and χ⁽⁶⁾ confirms to a very high degree of confidence previous conjectures about the location and strength of the singularities of the n-particle components of the susceptibility, up to a small set of 'additional' singularities. The exponents at all the singularities of the Fuchsian linear ODE of χ⁽⁵⁾ and the (as yet unknown) ODE of χ⁽⁶⁾ are given: they are all rational numbers. We find the presence of singularities at w = 1/2 for the linear ODE of χ⁽⁵⁾, and w² = 1/8 for the ODE of χ⁽⁶⁾, which are not singularities of the 'physical' χ⁽⁵⁾ and χ⁽⁶⁾, that is to say the series solutions of the ODE's which are analytic at w = 0. Furthermore, analysis of the long series for χ⁽⁵⁾ (and χ⁽⁶⁾) combined with the corresponding long series for the full susceptibility χ yields previously conjectured singularities in some χ⁽ⁿ⁾, n ≥ 7. The exponents at all these singularities are also seen to be rational numbers. We also present a mechanism of resummation of the logarithmic singularities of the χ⁽ⁿ⁾ leading to the known power-law critical behaviour occurring in the full χ, and perform a power spectrum analysis giving strong arguments in favour of the existence of a natural boundary for the full susceptibility χ.