Singlet oxygen actually has no unpaired electrons, while the normal O2 molecule, triplet O2, has two unpaired electrons in p antibonding orbitals when the molecule is in its ground state. The designations "singlet" and "triplet" are spectroscopist's terms. In a magnetic field, a singlet molecule (one that has all of its electrons paired) exhibits only a single energy state, while a triplet molecule (a diradical having two unpaired electrons) exhibits three energy states corresponding to the possible orientations of the electron spins with respect to the magnetic field (both spins "up", both spins "down", or one spin "up" and the other "down").
As for the Lewis structure of singlet O2, you've hit upon a weakness of Lewis structures: we can't even draw a proper Lewis structure for normal (triplet) O2! The usual Lewis structure of O2, O=O, with two lone pairs on each atom, is incorrect, because it does not show the two unpaired electrons that the molecule actually has. If we look at the molecule in terms of hybridized orbitals, with each oxygen atom sp2 hybridized, we still have the same problem, where do we put those two unpaired electrons? The only approach that correctly explains the presence of the upaired electrons in the O2 molecule is molecular orbital theory, which uses molecular wavefunctions to describe the configuration of the electrons in a molecule. Molecular orbital (MO) theory gives "bonding" and "antibonding" molecular orbitals, something that can't be shown with any dot structure. Here are crude MO energy level diagrams for triplet O2 and singlet O2. I'm using a "|" to represent the electrons, but these should really be up or down arrows (if you haven't seen any of this before this probably won't make much sense!). The asterisks (*) indicate the antibonding molecular orbitals, and energy increases as we go upward in the diagrams.
Triplet O2 (two unpaired electrons) Singlet O2 (no unpaired electrons)
(This is the ground state) (This is an excited state)
s2p* s2p*
| | ||
p2p* p2p* p2p* p2p*
|| || || ||
p2p p2p p2p p2p
|| ||
s2p s2p
|| ||
s2s* s2s*
|| ||
s2s s2s
The upper pair of electrons in singlet O2 can be in either one of the p2p* MOs since those orbitals have the same energy (are degenerate).
Most of the time, Lewis structures give us a reasonable picture of the molecule in terms of types of bonds (single, double, or triple) and the number of lone electron pairs on atoms. But we have to keep in mind that Lewis structures predate concepts such as hybridization, molecular orbitals, and even quantum mechanics. Here is a quote from one of my textbooks regarding the Lewis structure of O2 (with my comments in parentheses):
"Molecular orbital theory explains why O2 is paramagnetic, indicating two unpaired electrons, whereas the Lewis theory fails. The Lewis structure for O2 has no unpaired electrons: (dot structure shown). The only possible Lewis structure with a double bond and two unpaired electrons violate the symmetry of the molecule by making the oxygen atoms different, and make it appear that the unpaired electrons are both associated with a particular atom: (dot structures with the electrons in one lone pair on one of the oxygens unpaired as separated dots). You can partially redeem the Lewis structure by saying that these two structures are the two resonance structures for O2, and that the true structure is unrepresentable but has the character of both resonance structures in equal amounts. But this treatment hardly seems worth the effort. It is easier to abandon Lewis structures and to think in molecular orbital terms." (From Chemical Principles, 3rd Ed., by Dickerson, Gray, and Haight.)
Good question, hope my usual lengthy explanation is helpful!
Steve