Do frequency adverbs that go at the beginning of sentence need a comma? (Occasionally, Sometimes, etc.)

Solution 1:

As neither variant is confusing, I'd use either, depending on whether I thought a pause sounded natural / desirable.

I often go swimming, and I love fishing, clay-pigeon shooting, and bird-spotting. I often go dancing. Occasionally, I play football with my friends.

but

I do get a reasonable amount of exercise. Occasionally I play football with my friends, but my real love is swimming.

If you want an authority to sanction this,

Chicago Manual of Style 5.69:

"When [transitional adverbs] are used in such a way that there is no real break in continuity and no call for any pause in reading, commas should be omitted.

Transitional adverbs are traditionally even more attached to their following commas than adverbs of frequency.

Note that some frequency adverbs require inversion if fronted:

Seldom / Rarely / Never does the king now venture beyond his palace walls.

Solution 2:

Comma "rules" are tricky. Most are best thought of as guidelines than rules.

Setting of introductory elements is indeed something that most sets of such rules would say you need a comma, so "by the rules" the form with the comma is the correct one.

Conversely, if the occasionally was only the start of an introductory phrase then the same rule would have you not put a comma there, but after the rest of that phrase:

Occasionally I play football with my friends, but I'm not a physically active person.

But again, any such rule is best thought of as a guideline. If you want to emphasise by setting a word off with a comma or de-emphasise by omitting one, and the result is understandable and unambiguous, then go for it.