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Is there an advantage to using 91 octane vs. 89 when it comes to milage?

Not if your owner's manual doesn't recommend it.

Yes, to the oil company ($$). Some advocate running a tank of higher octane once in a while, believing it somehow cleans thing out. Some claim a slight increase in MPG, although this has never been my experience. I have run some of the higher octane when prepping for a tailpipe emissions test, hoping it would make my clunker more likely to pass, but even that was probably a waste of money. I'm told, and I believe, that it is best to use the lowest octane gas that will power your car without causing it to ping on normal acceleration (check this when the engine is at full operating temperature). Ad as Vooch wisely says, check what your manufacturer recommends.

Octane rating is a measure of the fuel's ability to resist detonation in high compression engines, nothing more.

Octane is not something that is added to gas to make it more powerful, it is a reference fuel against which gasoline's anti-detonation quality is compared. Pure octane (actually a particular isomer of it, 2,2,4-methyl pentane) has an octane rating of 100 by definition.

Burning a gallon of high octane fuel does not release any more BTUs of heat than regular gas does, if anything, the BTU/gallon of premium may even be lower.

If your car doesn't need premium, save your money.

Popular Mechanics weighs in on the octane issue here:

http://www.popularmechanics.com/automotive/how_to/4264939.html

I've been putting 87 Octane in my wife's car, but the manual recommends 89 over 87 for that particular engine, and also says absolutely NOT to put in 93. I wonder if the 3 or 4 percent increase in price (from 87 to 89) would be offset by higher gains in MPG? Anyone smart on this subject?

My best fuel economy tanks have run on 85 octane fuel. :)

My toyota engine (ZZ-series) have a rather high compression (10.5:1), and using higher octane fuel gives a lot of difference with regards to low-rpm driving. I can climb hills in 5'th gear @ 1300 rpm @ 20-30 MPG.

With a US type, oversized engine with automatic transmission, commuting at the highway, you would not gain anything.

My best fuel economy tanks have run on 85 octane fuel. :)

85 octane is only sold in high altitude areas where engines don't need high octane. It probably is the high altitude that gives you the better gas mileage. When the air only has 85% of the density of sea level air, the air resistance is also only 85% as much. Also, your engine effectively only has 85% of its sea level displacement if you measure displacement in pounds of air instead of cubic inches.

85 octane is only sold in high altitude areas where engines don't need high octane. It probably is the high altitude that gives you the better gas mileage. When the air only has 85% of the density of sea level air, the air resistance is also only 85% as much. Also, your engine effectively only has 85% of its sea level displacement if you measure displacement in pounds of air instead of cubic inches.
Yeah, here where I live, the common grades of gas are 85, 87, and 91.

Hmm... I wonder why the altitude isn't helping the fuel economy of most of those around me. I link to think I have something to do with my fuel economy--not just my altitude. Anyway, most of my driving is at lower speeds where air resistance isn't as much of a contributing factor.