Social housing schemes are very inefficient and, ironically, further entrap their participants in poverty. I can't speak for Germany, but in Australia most state social housing schemes are heavily oversubscribed, and so are rationed. A new participant can spend years on a waiting list before finally getting access to rent-subsidised social housing.
And once you're in your rent-subsidised government owned dwelling, you're tied to that location. If you're offered a higher paying job that would require you to move, chances are you wouldn't because you'd either:
(a) have to go to the back of the rationing queue, or
(b) lose access to social housing entirely
In other words, the effective marginal tax rate on additional earnings are well over 100% if relocation is required. Therefore, the rational financial decision is to stick with your lower paying job (or unemployment) to hold on to your rent-subsidy. Hence the social housing 'poverty trap'.
From a 'whole of economy' perspective it's even worse. In the aggregate, social housing impedes labour mobility, in turn lowering economic output and increasing the rate of structural unemployment/under-employment. Furthermore, it saps land market efficiency, as older social housing locations (that were previously on the outskirts of the CBD) are now in 'inner-city' areas (due to metropolitan expansion). As a result, land that could be put to much more productive use is tied up by social housing tenements.
Although the above is a bit of a simplification, in the final analysis social housing is a complete policy disaster that creates bad economic outcomes for both the financially disadvantaged and society overall.
It would be far better if social housing (and the land it is situated on) was sold on the private market (at market value) and the money put towards direct rent subsidies (that taper as income rises). Although this still causes higher effective marginal tax rates (due to payment tapering), the system can be designed so that EMTRs do not come close to 100% and there are actually positive incentives for those receiving rent subsidies to relocate if offered a higher paying job.