Choosing a polespear speartip has a lot more to do with personal taste than anything else. Certainly some tips are better in certain situations than others, but they are all being propelled by a relatively low powered weapon at short range. Ideally you want a tip that is light enough not to slow down the spear, has good penetration and adequate holding power. There are plenty of speartips that satisfy this criteria to varying degrees, with trial and error being the best way to find out what balance is right for you.
Multi-pronged tips have several advantages that make them a good choice for a polespear. For one, they increase your chances of hitting a fish - not by much, but often just enough. Also, with more tines, you have a greater chance of hitting something vital. Multi-pronged tips generally have their barbs very close to the point so you don't need to bury the tines very far to have full holding power. This is especially useful when hunting fish in holes or that are backed up against a rock wall. Penetration out the back side of the fish is impossible in these cases, so its best to sink as many barbs in as you can.
Multi-pronged tips have their disadvantages as well. They are often heavy, which cuts down on maneuverability and speed. Also the multiple barbs tend to ruin a lot of meat, so head shots are preferable.
The paralyzer is very popular for polespears. It works well for small to medium fish. The three tines are arranged in a triangle and diverge from each other. When you hit a fish, the tines separate slightly, which holds the fish on the spear with a side-load. Although some people say it defeats the purpose of the paralyzer, you can also get them with barbs for extra holding power. Barbless they ruin less meat, barbed they hold better, its tough to decide.
Paralyzers generally have 3/16" diameter tines and mount to a rather heavy 6mm adapter. You can get slightly heavier 7/32" tines, but smaller ones are rare.
The tines are slender so you don't end up ruining much meat. However, they aren't rigid, and often bending where they meet the adapter. This flexibility can also take away from penetration. I once tried to spear a bass directly from the front and the paralyzer bounced off its head with absolutely no damage done. This was probably due to the diverging tines not sinking any points in.
Known as frog-spears, rakes, gigs, 5-prongs 3-prongs, and by an assortment of other names, these spearheads hold more than one point on a single plane. In the United States the most popular are the three-tined and five-tined versions. These tips are usually very rigid, which means good penetration. They hold better in bone than in flesh, so a head shot is ideal.
For the budget-minded spearo, you can pick up a 5-prong "Frog Spear" at K-Mart (or similar department store), mount it on a broom handle and have a simple and effective spear for $5. The quality of these frog-spears isn't great, but the price is attractive.
My favorite tip is the uni-planar 3-prong made by Trident (this is not the trident speartip pictured below). It has four rather large fixed barbs, the center tine carries two, the outer tines carry one each. I think it has equal holding power to a typical 5-prong gig, but is faster and has better penetration. There are other 3-prong heads that are not nearly the equal of the Trident brand one, so don't think they are all made the same.
I'm not sure why these are called "tridents" because every artist's rendering I saw of King Trident showed him with a uni-planar 3-prong. Anyway, these tridents are like a short paralyzer with fixed barbs on stouter tines that do not steadily diverge. I have seen them available in two sizes. The larger was probably almost the equal of a good Trident (the company) 3-prong, but lacking one fixed barb and being slightly heavier. The smaller one might be good for little fish if you shoot them in the head.
This is the tip that came on my brother's small Beauchat speargun he bought in France. It probably works well enough on a small gun. I have never tried one of these tips on a polespear as it seems to lack the positive features of either the paralyzer or a good 3-prong. Sort of a middle of the road choice, I guess.
Speartips with a single point are the most popular style for spearguns, but many of them work fine on a polespear as well. Speartips that have a point ground directly onto the shaft are often called Hawiian or Tahitian shafts. These points are lighter than a threaded tip and often a better choice for a polespear. However, one good reason for having a threaded tip is that its easier to get the fish off - just unscrew the tip.
In general, tips that are sharpened to a round point are better around rocks, whereas a tri-cut tip will provide better penetration, at the cost of durability.
Single floppers hold one hinged barb about 3" from the speartip. A good single flopper on a light and stiff spear is about as good a combination you can get. The flopper has adequate holding power and the spear has better penetration than anything else. I have seen them in diameters of 1/4" to 5/16", and I like the lighter ones. The only drawback you could weigh on this tip is that because its a single point you have less chance of hitting something vital. Also, they aren't great for shooting fish up sitting up against a rock as the tip must penetrate out the other side for the barb to engage.
Dual floppers are the most popular type of screw-on tips for speargun shafts. Usually these tips are larger and require more power than you can get with a polespear, hence they lack good penetration. However, there are some light speargun tips which work well on a polespear.
Polespear slip-tips are for big game. I think this design was pioneered by the Eskimos, though it may have shown up in other cultures as well. Eskimos could take Walrus with a slip-tip and a lance.
Nothing has more holding power than a slip tip; because the tip isn't rigidly fastened to the spear, the fish can't use the spear as leverage to work its way off the tip. Once the tip is through the other side of the fish (or under the skin of an Elephant Seal) it turns sideways and becomes a flat barb with great holding power.
One drawback of the slip-tip is that its entire length must pass through the fish so that it can neatly toggle on the other side. The better the penetration a slip-tip has the less this is an issue. Naturally, slim slip-tips with a tri-cut tip are the best. Another drawback of the slip-tip is that you lose control over the fish. You can no longer steer the fish around as well.
Sliptips for spearguns don't have a long shaft that the tip sits on. If your polespear has a shaft at least 6" long with threads on it you can use a sliptip designed for a speargun (like the one pictured). Otherwise you will need a sliptip that is designed for a polespear. Both JBL and Biller make fine polespear slip-tips.