How are registers implemented in microprocessors?
Option 2. Here is the most basic gate-based semiconductor memory, the R-S flip flop:



When either R or S is triggered, the feedback loop causes the system to maintain its state until the other input is triggered. As such, it is capable of storing a single bit. Static memories (of which registers are just small slices of) are composed of arrays of thousands to millions of similar circuits.

Dynamic memories, OTOH, do use capacitors to store bits. Since the capacitors are very leaky they must be refreshed very often, and this leads to higher power consumption and lower access speeds than with static memories. This is balanced by much higher density, since the circuitry is much simpler.