Recent discovery that single-stranded DNA (ssDNA) binds to carbon nanotubes with high affinity to form soluble hybrids has received great attention as a promising approach to solving the long-standing problem of nanotube solubilization and separation. The mechanism of this process, including the nature of the DNA-nanotube interactions and the molecular structure of the hybrids is still not well understood. Here, we use all-atom replica-exchange molecular dynamics simulations to study the association of several ssDNA decamers with single-walled carbon nanotubes of different chirality in an aqueous environment. The oligonucleotides are found to readily adsorb onto the nanotube surface, after which they undergo a slow structural rearrangement. Cluster analysis of bound DNA conformations as well as population distribution maps computed as a function of several local and global order parameters show that the hybrids exhibit a complex morphology with DNA strands assuming a number of distinct backbone geometries, which depend on both DNA sequence and nanotube diameter. In contrast, the nucleotide bases are found to align parallel to the nanotube surface with a high degree of orientational order. While the binding appears to be primarily driven by energetically favorable pi-stacking of DNA bases onto the nanotube surface, equilibrium distribution of hybrid conformations is modulated by a complex interplay of forces, including the DNA conformational strain and solvent interactions. As a result, the hybrid free-energy landscapes are found to be rugged, with multiple low-lying minima separated by high barriers, several of which are significantly populated at room temperature. Qualitative differences are observed in free energy profiles of purine- and pyrimidine-based oligonucleotide sequences and are attributed to the difference in self-stacking propensity of the bases.