I suspect "shady circumstances" is caused by a simple misunderstanding of the type home posted. Those with Notice of Default or even for sale have missed a payment or have been reported sold from Sheriff Auction (but not completed the redemption period) so most are not available for sale. This information is provided by a 3rd party provider. You can choose to knock on doors until the correct home is located and try to negotiate with the homeowner. Because there are no addresses, most people shy away from this choice, but some make money using this method.
I would agree with Maureen, that every Michigan agent on this website would have a search that updates daily for your area. These sites with indicate IDX next to the homes (Internet Data Exchange). We all pay additional money for this to be available on our sites because of consumer demand.
The local MLS allows agents several days to report a home as pending sale or sold. Even if the agent were to report the sale immediately, there would still be a delay in removing the listing on any website.
As to multiple offers, your best method to receive information about new properties is to have listings emailed by ONE agent. They can set the search to only send homes new on the market (last 14 days). Keep in mind, the bank is still trying to get the most for each home and will wait until several offers have been submitted before even looking at any of the offers. This may happen even if you were to offer more than asking price for the home.
A buyers agent, would be able to represent your best interests.