These are my study:
It seems like the verb "cycle" refers to "go by bicycle" is often used in British English more.
And the verb "bike" can be used for both "bicycle" and "motorbike"
"a bike" can refer to a bicycle or informally refer to a motorcycle
But I don't know when we say "I cycle to work", "I bike to work" and "I ride a bike to work".
"Cycle to work" seems the most likely. "Bike to work" could mean bicycle or motorbike. "Ride a bicycle to work" is exactly the same meaning as "cycle to work, only slightly longer".
There is a government grant scheme called Cycle to Work
But all three sentences are correct and could be used by a cyclist to describe how they get to work.
I suspect there are differences depending on what side of the Atlantic you're on, but for me, a veteran cyclist from Canada, all three of your sentences are correct and natural to describe commuting by bicycle.
"Cycle" and "ride a bike" only refer to bicycles. I was unsure about "bike" as a verb, so I asked my friends who ride both motorcycles and bicycles, and they say only use "bike" as a verb to mean ride a bicycle, never a motorcycle.
There may well be differences between American and British English, or between sub-regions in either.
Where I live -- Michigan, USA -- we would typically say, "I ride a bicycle to work." If you said "I cycle to work", people would know what you meant, but it's not something we say here.
"Bike" can refer to a bicycle or a motorcycle, but, at least among people I talk to, it more commonly means bicycle. If you said, "I bike to work", I'd understand you to mean that you ride a bicycle.
But it could mean "motorcycle" in the right context. Like if we were talking about motorcycles, I'd assume you meant a motorcycle. Like if you said, "My bike is a Harley", I'd take you to mean motorcycle because I've never heard of a Harley bicycle. (Is there such a thing? I suppose it's possible.)
The question has been muddied because it brings in regional usages. Let's set those aside, since it's pretty simple: cycle as verb and bike as verb are British and American respectively.
But let's focus on the question: Do people say "I [verb] to work," or "I [verb an object] to work." The short answer is: it doesn't much matter. Both are reasonable.
The longer answer: We've got an unusual situation with bicycles here. There's no similar verb form for riding a horse, say; you can't "horse to work." And on the other hand there are other things you could "ride," like scooters or skateboards, so "I ride to work" is vague.
"I ride a bike to work" is not awkward or non-idiomatic. It might be that people would be more likely, though, to say "I ride my bike to work." But this construction is no more or less common than "I bike to work," and there's no significant contextual reason to choose one or the other.
