I'd suggest you try anyway. Two options:
1) The easiest option is to see if when you check in with UA you can get the bags tagged all the way to your final destination on MI. If it works, you're all set. (It is possible for two separate tickets to be "joined" for this purpose provided there is no administrative issue preventing this.)
2) The second option assumes you were not able to interline the baggage transfer during check in with UA. In this case, the transfer may be possible in Singapore. What you do upon arrival at SIN is to head straight to the Transfer Desk. This is airside so don't go through passport control/immigration. At the Transfer Desk, get yourself checked into the MI flight, and produce your UA luggage claim check. SIN luggage handling would locate your bag and send it to MI. You can then just go straight to your gate never having gone to ground side. Both UA and MI are in the same terminal so your walk to the next gate will be straightforward.
http://www.changiairport.com/en/pass...e/transit.html