As far as I can tell 0.4 introduces heavy changes to the entire LSPDFR system, namely the new crime system.
I am under the impression that this will break compatibility with most 0.3 mods. I would imagine that the only mods that might work are the very simple ones, such as mods that calm peds around you or mods that lets you place basic objects into the game world. Anything more complicated than that would most likely require an update or even a complete rewrite.
I'm rather new in this community (3 days old in fact...), so I don't know if this is the case, but the LSPDFR team might release API documentation ahead of time, so that modders can start updating ahead of time.
From my limited understanding, I believe that 0.4 provides a more robust underlying system to manage and control crimes and so-called crime events. Most of the stuff that mods had to take care of themselves are now being taken care of by LSPDFR, so that mods have a better way of exchanging information. I am so excited it hurts.
I would recommend starting fresh with 0.4, and see which mods are getting updated from there.
Cosmetic mods should be fine. Cars and clothes should still work, with few exceptions.
It's stuff like callout plugins and heavy mods, such as LSPDFR+, Computer+, TrafficPolicer, Arrest manager etc. that is most likely to completely break.