The two first katas of shotokan karate.

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A short review of how the Fairbairn method works with some different models of knives

The knife used in the video is the british army dagger known as the Fairbairn-Sykes. This is not something you are likely to carry around with you for no reason, as its use is optimized for combat. If you just want to practice the method you may use a more everyday knife available to you. This is also a more realistic stance if you mean to use it for self defense.
I picked out some common types of knives and tried how well they work in regards to the movements and the grip. The methods I used were technical practice, shadow fighting, and application on a dummy made from several layers of industrial quality rubber mat.

The blue knife:
Works well for all movements, light and easy to handle. What you need to keep in mind is that upwards slashes require a change of grip as this knife is single edged (same for all knives I tested), not double like the dagger. The plastic handle will also get slippery if wet. If you only mean to use it for shadow fighthing, this works well. It's also visually intimidating, which should play a role in a real life sitation. Only showing this to someone could repel a less motivated attacker.

Orange folding knife:
The grip works perfectly as shown in the instruction. Easy to hold between thumb and index finger. Downside is that it is worthless for stabbing. The blade is too broad to penetrate more than a few centimeters unless using considerable force, the opener tool makes it easily stuck if you do manage to penetrate the material. Works well for practicing the motions, but less usable in a real life situation as the number of techniques are limited to cutting. However, it's more likely that you will carry a folding knife with you, so from this perspective it may be better than other alternatives even if your range of movements are limited.

Metal folding knife:
The one I used has a smooth handle, but otherwise similar.
The grip for this type of knife has to be turned 90 degrees because the handle is too thin to be held as shown. Holding it between thumb and index finger works very well. But you are limited to inward slashes and stabbing below shoulder level. Outward slashes simply don't work as your arm does not function well at this angle. It may look like this knife is difficult to use and easy to drop, but when testing it on the dummy it worked surprisingly well for stabbing.
It can be argued that this type of knife does not look particulary threatening, which is bad if you want to repel someone by showing it. If your intention is to just carry it and use it, it is probably a good alternative.