Random thoughts on this subject:

Competitive pressures seem to have at least some effect. Both NW and UA fly (their own aircraft all the way) from the U.S. to SIN, both currently have a six-day miniumum stay. It seems that if one drops the requirement the other one will.

Six-day minimum stays to SIN have been around on UA for ~ 12 months. These are part of the fare rules for HQVSLKT fare classes. You have to buy an M fare or higher, ~ $1,500, where an H fare might be ~ $900 and a Q might be $750 (I've taken some liberties averaging East and West coast departures).

Occasionally UA drops this requirement to either a Sat. stay or a 4 day minimum stay. They also offer weekly e-fares that have a four day min. stay.

A business trip from the U.S. to Singapore is not hard to stretch out to six days considering it takes three to fly out and back, add in a weekend and your good to go.

Many low priced economy fares on UA have advanced purchase requirements that restrict some business travel purchase. (Some Corp. contracts exclude low priced fares from the discount schedule and aggregate revenue thresholds.)

I assume that six day min stays are primarily meant to maximize revenue by steering business travelers, to higher fares. I really don't think mileage runners are the target?