Looking for a track ring and I see these two bcds offered.

Does the 130 mm bcd provide more leverage or some other advantage over a 110 mm bcd chainring?

Do you already have a crank? If so get the one that matches the BCD of your crank. If you are starting from scratch, you will find different ranges on sizes more common in one standard or the other. There is also 144BCD.

130mm offers less flex.

Do you have a crankset yet? Is this for a street fixie/track or for a mountain single speed?

The advantrage is that 130bcd rings are everywhere since they are road standard. 110 bcd is garbage.

Garbage? How do you figure?

Because it's hard to find rings in any shop?

Where are you going to find a 1/8 110 bcd ring? Eh?

I don't bother with 1/8". 3/32" 110bcd rings are pretty ubiquitous, and are available all the way from 33t to 60t or so.

110 is a fairly popular size actually. Especially since the compact crank revolution lately. If you ever want to run as low as a 34T chainring, up to a 50T, then go with 110. If you only want big rings up front, go for the 130.
This is how I've always set up my track bike, normal chain rings work fine.

Normal chainrings may work for the average joe, but 1/8 rings flex less and seem to hold a better chainline (imo) on real track bikes, plus iirc chainring bolts sometimes bottom out on the thinner rings, requiring washers to make them work.

If they're posting on bikeforums asking for equipment advice, it's a pretty good bet that they're 'the average joe' and not some monster-thighed track crusher.

What is the advantage of a 3/32 over a 1/8" chain?

Thicker chain is less likely to break, but this is only really applicable with elite and pro track sprinters with massive thighs who put a lot of force on their drivetrain. Most people wouldn't care, but that's the reason single speeds come with 1/8 chains. They will work with 3/32 chainrings and cogs, which explains the argument above on why it shouldn't matter.

Most have bushings between the rollers and plates so it's slightly smoother. They tend to wear longer because more surface area is contacted. Also, they're pretty ubiquitous for track, so you could swap parts easier on the 'drome. Disadvantages include running louder and slightly more weight.

Although I don't have monster thighs (but they're decent, IMHO), I still prefere 1/8" drivetrains, because practice has shown that the 1/8" chainring will outlast several times the 3/32" chainring of similar size and equal material.

1/8" cogs also seem to last longer. This is harder for me to tell exactly, because I haven't paid as much attention as with the chainrings, but the ratio seems about the same.

I can say that a 1/8" chainring can probably outlive 3 3/32" chainrings subjected to the same force and stresses. This is purely my own experience. YMMV.

EDIT: and no, I don't have a clear scientific explanation for this. Such a huge difference is hard to explain.

Sugino used to make 1/8" chainrings with both 110 and 130 mm BCD holes. God bless them for that.

Noone else, as far as I know.