All conventional plastics marked with the 1 to 6 resin identification codes are recyclable. Some types are more easy to recycle than others, for example recycling PVC is more difficult than PET. Also very little polystyrene (PS) is recycled simply because it is not cost-effective to do so.
The majority of conventional plastics is downcycled into a lower grade plastic and as a result plastics are usually only recycled once.
Most bioplastics are compostable plastics and I've heard mixed stories about their recyclability. Some claim that compostable plastics cannot be recyclable because this quality interferes with the plastic recycling progress. Others say that a compostable plastic like PLA for example can be recycled, at least in theory. In any case different plastic types need to be recycled separately. If you throw a compostable plastic in a batch of conventional plastics, it just acts as a contaminant and lowers the quality of the recycled plastic. You need a large and constant supply of a particular plastic type before recycling becomes cost-effective. So far I've not heard of a recycling facility that actually does recycle a compostable plastic (not counting composting or thermal recycling).
Contrary to popular belief compostable plastics like PLA are not compostable in your home compost heap (see also Which compostable plastic types can be composted at home?). PLA can be composted industrially, but it seems very little recycling companies do this and most compostable plastics are incinerated (see also How is compostable plastic recycled/composted industrially?)
The Wikipedia article on bioplastics mentions one type of bioplastic that is recyclable which is Bio-derived polyethylene. This is exactly the same material as traditional PE made from fossil-fuels, but in this case it's made from ethanol.
Some people say that OXO-biodegradable plastic is a recyclable bioplastic, but this specific type of plastic is a bit controversial. It's made from conventional polymers with some added salts that catalize the degradation process, so it isn't really a bioplastic. The manufacturer claims that oxo-biodegradable plastics do not just fall apart into tiny plastic pieces and are really biodegradable. However so far no independent research has confirmed this. For more information on this see also Do oxo-degradable plastics really biodegrade? and Mexico’s PE suppliers denounce oxo-biodegradables.