When should you use a semicolon *with* a conjunction?

I know the basics of a semicolon—at least I think I do. Aside from delimiting verbose lists, it separates independent clauses of a sentence. So, if you have two independent clauses in a sentence, you can either separate them with a semicolon, or a comma along with a conjunction—like "but".

However, I've noticed a few authors actually using a semicolon with a conjunction, like:

<independent clause 1>; but <independent clause 2>

Can anyone shed some light on when this is preferable to just a comma? Is this simply a matter of personal preference?

To me, it seems to be purely personal preference. The semicolon between clauses suggests a connection between the sentences that is stronger than if there were a period between the two.

As (to me) it is generally acceptable to start sentences with the short conjunctions and and but, I believe the general rule can extend to independent clauses joined by a semicolon.

Possibly:

He is the most disagreeable person I've ever had the misfortune to meet, and I dislike his style; but I must admit that he gets the job done.

You can use a conjunction whenever you feel like doing so but it is more common when the semi-colon isn't able to distinguish between potential options:

I like vanilla and chocolate; strawberry is okay.

I like vanilla and chocolate; and strawberry is okay.

I like vanilla and chocolate; but strawberry is okay.

Of these, I prefer the first and third. The inclusion of the semi-colon splits the sentence appropriately and the conjunction helps clarify the tonal shift of the last segment.

Since some people frown on beginning sentences with conjunctions, it's one of those rules you have to break somewhat carefully. Let's take MrHen's excellent example above:

I like vanilla and chocolate; but strawberry is okay.

In this case, a comma would probably get the job done. The semicolon offers more of a break, and here works to distance strawberry a bit more - the author is tossing a backhanded sop to strawberry. The two clauses are so tightly linked in meaning that even though we're starting the second one with a conjunction, the reader doesn't notice.

My observation is that semicolons are most commonly used with conjunctions when they're joining two sentences that themselves have internal punctuation (commas). The semicolon helps clarify the 'weighting' of the different parts of the compound sentence. The same rationale for using semicolons in lists where some components of the list have commas. An example:

When they went down to the beach, wind was gusting in off the water; and on the horizon, clouds steepled up high into the glowering, grey sky.