How to check what status sent by SCCM of specific request.
How to check what status is sent (Installed/Failed) by SCCM of a specific request.
Which logs show the status that SCCM has given for specific software installation.
Usually, the best place to look is in the request log for the request. This log gives you a complete overview of the request lifecycle. This assumes that you have verbose logging enabled, and also have "Enable request life cycle logging (Use only for troubleshooting)" enabled under settings->website. The request log files are are located in the logs/requestLog folder. Other log files which contain information about the status check are StatusUpdate.log, and DeploymentStatus.log. These will typically contain the status queries being run.
If you just want to view the status of a specific request, you can go to My Requests and click on the CD-ROM icon in the Status column for that request. It should bring up a small pop-up with the detailed response from SCCM (e.g. Requirements Not Met).
Edit: I guess I should also note that if you don't configure your catalog items to clean up the collection membership when finished, you can also view the status or run reports directly in SCCM.