I am a Singapore citizen currently living and working in the UK under a UK spouse visa but my wife's company plans to relocate her to Germany. Would I have to apply for a German visa to be able to work in Germany? Or is that not necessary due to the freedom of movement law in the EU?
No, being a UK resident does not grant you the right to work in (or reside or even simply visit) Germany (see also Does a non-EU citizen residing in one EU country need to apply for a visa to move to another EU country for work?). The EU freedom of movement applies mostly to EU citizens and only indirectly to non-EU citizens. So your options to join your wife in Germany will depend on her status and citizenship:
If your wife is a British citizen, her freedom of movement rights will grant you the right to work in Germany as soon as she works there (or otherwise makes use of her treaty rights). You will however need to apply for a German residence card, which should be free of charge.
If your wife is not a British citizen, EU law does not generally apply but it should still be possible to get a spouse visa in Germany, under the conditions set by German law (within the boundaries defined by directive 2003/86/EC).
The difference is important because there are complex rules and additional requirements for partners of German citizens or third-country (i.e. not EU) nationals. In this case, you will need to obtain a visa before entering the country (as a citizen of Singapore, you can visit for a short time without visa but you are generally not allowed to apply for a residence title from within Germany). I think that whether you will have the right to work could depend on your sponsor's status but if your wife comes to Germany to work, then you should be allowed to work too.
(I assume your wife is not a German citizen nor from another EU country than the UK, because if that was the case you would presumably hold an EEA family permit or UK residence card instead of a spouse visa.)
Finally, directive 2003/109/EC grants some additional rights to permanent residents in the EU. I don't know if it applies to your situation but in any case, it's only supposed to facilitate the process and does not entail a full equivalence between residence permits of different member states.