Do you have information about the cleanup of single-stranded DNA (ssDNA) with QIAquick columns?
FAQ ID -759

As a rule of thumb, single-stranded DNA binds to silica with approximately half the affinity of a double-stranded DNA fragment of the same length under the buffer conditions used in the QIAquick and MinElute Kits. Even though no systematic experimental data exists, we expect that recovery of ssDNA fragments of approximately 200 nucleotides and below will not be very efficient after cleanup using the QIAquick PCR Purification Kit or MinElute PCR Purification Kit. By comparison, it should be possible to purify fragments longer than 140 nucleotides using the QIAquick Gel Extraction Kit.

Note that recovery of single strand DNA is influenced to some degree also by factors such as base composition and secondary structure. It has to be determined empirically by the researcher if cleanup of single-stranded DNA with QIAquick columns yields satisfactory results.