The logo shown depends on the marketing carrier of an itinerary. If an itinerary is composed of multiple legs (segments), the logo will be shown when all legs have the same marketing carrier; otherwise, a string with the names of the carriers will be shown, e.g "Singapore Airlines + SilkAir".
To better explain this, let's walk through the following examples:
a) Flight TPE-DPS via SIN with two legs, the first leg has both marketing and operating carrier as SQ; the 2nd leg has marketing carrier as SQ but operating carrier as MI.
In this case both legs and the itinerary will be branded as SQ, but annotations like "Partly Operated by SilkAir" and "Operated by SilkAir" will be added for clarification.
b) Flight TPE-DPS via SIN with two legs, both of the legs have their respective marketing and operating carrier, SQ and MI.
This results the two legs being branded to their respective brand names, and the itinerary treated as a flight combination - "Singapore Airlines + SilkAir".