Update: I called Singapore Air's US toll free number. The agent I spoke to said that I need to fill out a consent form in order for them to share my information. The form was on their website. I filled it out and emailed it to the address he gave me. About 12 hours later I received an email from singaporeair saying that my itinerary had been emailed. It was, complete with the booking reference and e-ticket number.

Another interesting thing is that the credit card transaction updated to show a number in the description, part of which seems to be related to the e-ticket number. Also, the price dropped by $1.43. The reservation was booked in INR (Indian Rupee). I'm guessing that the exchange rate changed between the time I made the reservation and when it was confirmed.

The agent told me that the reservation probably didn't show up automatically in my Krisflyer account because I entered my United MileagePlus number for accrual during booking.

Thank you for your advice to call Singapore Air.