2010 isn't far back in history enough to be archived, you should still be able to request it directly from the police department, should be no problem as long as you are a party or attorney connected to the report. The procedures in which to obtain a police report can be checked on their records department webpage. Without knowing which specific department made out the report, that's as specific as I can get. If you know the area, post it and someone else may have put in a request for the same thing.
When you click on this link http://ujs.sd.gov/ you will be taken to the state of South Dakota's unified judicial system. There will be a box that says "Court Records", that will take you to a search page and see how to get criminal records (goes back to 1989), civil going to 2003 and civil money judgments. There are other self help services from the state.
Most recent arrests can be found in individual police, sheriff or any law enforcement agency involved. They will have a webpage showing their arrests in local jails, Pennington is a good example, you can see arrests and charges by offender's name. If you have more information about where and names, you can look up the courthouse records, which are generally considered public records unless sealed by a judge or otherwise.
Although all background checks are not the same and can't speak for every one of them since there are so many. Background checks they will not include specific details about an arrest if at all. First, the arrest do not show up unless there's a conviction following. Police reports are produced by the agency which processed the arrest.