Some of my LiPo batteries got "puffy" after a few months. Is it still safe to fly them, or should I safely dispose of them?
You should dispose of them.
A LiPo battery has three parts: the anode (negative plate), the cathode (positive plate), and electrolyte (sandwiched in between the two plates). Electricity is formed by electrons moving from the cathode to the anode through the electrolyte (which helps the electrons flow).
The electrolyte decomposes over time, which results in the formation of elemental lithium and oxygen. Some of it combines into lithium oxide and sticks to the anode and cathode, but some of the oxygen remains a gas. This gas is what causes the puffiness you see.
Oxygen likes to burn, so if you use a puffy battery, the likelihood that the moving electrons will ignite the oxygen is much higher, so once it becomes puffy, you should stop using it and dispose of it.
Puffed batteries aren't going to fly well anyway. Usually, they will suffer from pretty bad voltage sag which will most likely trigger your copter's low voltage alarm and possibly auto-shutdown early. So, ultimately flight time will suffer and on these tiny packs, flight time is already low anyway. There's also a risk of bursting or catching fire if they rupture. Not really worth it as these tiny lipos are kind of cheap and expendable.
Not sure if this applies to your problem or not but... If you want them to last longer, don't store them charged. Most small quads will shut down right around the battery's storage voltage so just leave them until you want to fly again, then charge them just before use. Charged lipos deteriorate much more quickly if they are stored fully charged.