This comes down to whether Singapore Airlines is able to check your bags through despite the two separate tickets. They are not required to do so under Star Alliance rules, but [they're generally known to be on the more generous side][1], compared to some airlines that will absolutely refuse to do it. Make sure you have all the ticket information for your Lufthansa flight, present it to the Singapore agent at check-in, and ask nicely if your bags can be checked through. If the agent is unsure how to do it, perhaps ask if there is someone else who can help.

If they cannot check your bags through, you'll have to go through immigration, pick them up in Munich, take them through customs, check them in at the counter, then go through airport security. This shouldn't be a huge problem though, as you have a fair amount of time if your flight is not significantly delayed. For this itinerary, you'll have to go through Schengen immigration and airport security in Munich whether you need to recheck bags or not, so these are largely steps you'd be doing anyway, just not in the transit area; the only additional steps are the stop at baggage claim and the ticket counter, plus the walks in between. 


  [1]: https://www.flyertalk.com/forum/singapore-airlines-krisflyer/1500463-singapore-policy-interlining-separate-tickets-2.html