I can make the dirtiest systainer look like new. Done it many times.
First I put it under the shower, and clean with normal soap while the shower runs. I use a brush for the dishes, then a sanding sponge and finally a wire brush for pans. The last one is really good for tough dirt lodged into the texture.
This cleans up the systainer quite a bit but is not good for the stickier dirt. So after the systainer has dried I now use a cloth with an ammonia solution for the tougher dirt and treat the systainer with that. Thicker dirt is scraped off with a chissel or a sharp putty knife.
Rarely I am done now, now we go to the stronger solvents. First white spirit. Again I wipe it, now with a cloth drenched in white spirit and this will get off the dirt that doesn't dissolve in water. Also very good for removing stickers and their glue.
But most of the time that isn't enough either. Those systainers can get very dirty when they're used by professional tradesmen, who use glue, paint, putty and caulks.
All methods above were non-destructive, but now we have to resort to tougher measures to get the really, really heavy dirt off.
Paint thinner or acetone will get most of your systainer clean as a whistle. They are very strong solvents, so be carefull as they can also dissolve your systainer itself. But if you use it only superficially on the dirty spots, by whiping with a cloth, there's not much damage done, maybe you only whipe away the texture so the systainers looks smooth on spots. Looks better IMHO than a big spot of paint. Thinner can also get rid of severe yellowing of the ABS.
In some spots the dirt is really heard to remove. I scrape even harder with a chissel, and I also whip out the Deltex sander sometimes to hit a spot.
In the end it can take quite a bit of work, but I always end up with an almost spotless systainer.