The excellent thread by callen on certamen fighting styles (https://forum.atlas-games.com/t/certamen-styles-clarifications-sought/5400/2). Basically, mechanics that provide you an advantage if you "land a blow and decline to do damage" require you not only to beat the defense total as callen assumes, but the weakening total too. This makes said mechanics less applicable (and thus the styles offering them less useful) and resistance more important (and thus styles that boost it or damage the opponent's more useful). In the light of this, of some very insightful comments in the above thread, and of my own experiences in the last few sessions of my saga, I'd try my own version of the "great, good, ok, poor, bad" classification of the various styles provided by callen. Comments are highly welcome. I'll just consider dedicated followers of those styles, leaving out those styles that have no dedicated followers (like sagittarius or bone-biter).
In a nutshell, the best styles seem either Provocator or Hoplomachus (very similar, with the former being slightly better overall); or, for hard-core duelists, Laquerius and Velitus (again, very similar, with the former being slightly better for dedicated duelists). Andabatus and Scissor are also good choices for magi with highly asymmetrical Arts and/or with ample vis to spend (the former with high Finesse, the latter with low Finesse). Essedarius and Retiarius are worthwhile under limited circumstances, but probably not worth choosing; other styles are almost certainly not worth choosing. More in detail:
Gladiator: BAD. The plain old "vanilla" certamen style from the core rules can be reproduced, at least mechanically, by almost every other style, even when not fighting as dedicated follower of that style. Thus it makes no sense to choose it from a player's point of view, in terms of sheer efficiency (of course, there might be other reasons).
Andabatus: GOOD. In any given round you can claim a bonus of your choice to your Attack total, up to 2xFinesse, in exchange for an identical penalty to your Defense Total. While the net gain is 0, if you win initiative you are going to strike very hard (inflicting fatigue penalties or even unconsciousness) before your opponent strikes you hard. Andabatus is thus particularly good if you have high finesse (and decent quickness), even more so if you have a high disparity in your two dueling arts and/or if you intend to use vis. Because it tends to make duels shorter, it tends to make botches rarer and favours the stronger duelist.
Bestiarius: OK verging on POOR. Only usable by Bjornaer, it provides a very marginal advantage in a narrow set of circumstances (you can substitute your Heartbeast score for Finesse score in a Muto certamen -- the latter is rarely significantly lower, and it only influences initiative anyways) and a truly great advantage in an even narrower set of circumstances that the savvy opponent will almost always avoid (add twice the Heartbeast score to Resistance in a Muto Animal or Muto Corpus duel).
Essedarius: OK, verging on GOOD. With Essedarius you can forgo causing fatigue on any attack that would have caused it, and instead force the opponent to make a Brave check or lose concentration, with an Ease factor equal to 3+(2xFatigue levels forgone throughout the duel). This sounds great in theory. If you forego causing 1 fatigue level for three times in a row, an opponent faces three checks, one at 5+, one at 7+, and one at 9+. The probability that an opponent without any bonuses or penalties will pass all three checks is less than 3.8%. In practice, it's not quite as good against a prepared opponent. First and foremost, a magus benefits from his familiar's silver cord, possibly from his Mentem form bonus (this is subject to interpretation, but I'd say it does), can spend confidence to fix a poor roll; last but not least, he can use magic to make himself braver. Essedarius is then a very strong style against casual opponents, but against opponents who have spent even a modicum of effort towards preparing themselves against the style Essedarius is almost as bad as a plain Gladiator.
Gladiatrix: POOR. Basically, you can sacrifice a successful attack to be able to gain insight on how "valuable" the other party considers a given concept, idea, good etc. This is useless for certamen itself. And it seems you could gain the same information by simply talking to the target and making a good Folk Ken roll.
Hoplomachus: GOOD, verging on GREAT. You get a bonus of your choice, up to 2xFinesse, to your Attack total, and take half of the bonus as a penalty to your Attack total. You also have the option of foregoing attack in exchange for a bonus of 3xFinesse to your Defense, but giving an opponent a free shot at you, albeit at a substantial penalty, seems useful only in some very, very specific circumstances -- e.g. if you somehow know/guess that your opponent is about to attack you with a lot of vis. However, you always lose initiative; Hoplomachus is then the only style that cannot duplicate Gladiator and is strictly worse if your Finesse is 0. It also tends to make duels long, grueling affairs fought to first botch, which favours the weaker party.
Laquerius: GREAT. If in a given round you accept a penalty to your defense (up to your Finesse) and land a blow declining to do damage, you can force that same penalty on your opponent for the rest of the duel. This would be a phenomenally good style if magi had, say, a dozen fatigue levels or more, because you could just take on some "damage" early on in the duel, in exchange for a hefty penalty to your opponent forever after. With five fatigue levels before incapacitation, it's not very easy to recoup that initial investment, particularly against the stronger styles such as Andabatus, Hoplomachus, Provocator, Scissor and Velitus. But if your certamens are munchkinistic affairs involving significant energy-boosting magic and/or theriacs this might be the strongest style of all.
Provocator: GREAT. This defensive style is similar to Hoplomachus; the differences are subtle, but generally weigh in favour of Provocator. First of all, a Hoplomachus always gives up initiative except against another Hoplomachus; a Provocator only takes penalty to it, which can be made arbitrarily small by accepting a proportionately lesser bonus. Second, Provocator gains a Resistance bonus up to his Finesse; the Hoplomachus gains a Defense bonus up to twice his Finesse, but takes half of that as a penalty to his Attack. While in both cases the net bonus can be as high as the mage's Finesse, to claim that bonus the Hoplomachus must take a more extreme defensive stance, thus sacrificing flexibility. Also, while it seems that a Defense bonus is superior to a Resistance bonus (because if you have a sufficiently high Defense, who cares about Resistance?) in practice, that's not the case: in my experience Stamina+Parma tends to match or surpass Intelligence+Penetration, and while a Defense roll can botch (thus negating any Defense bonus) a Resistance total cannot botch since it's not rolled. I ran some computer simulations, and indeed a Provocator appears to defeat an "evenly matched" Hoplomachus more often than not.
Pumilius: POOR. Basically, you can sacrifice a successful attack for the chance of inflicting a successful attack. If it's a very good chance, and you are likely to cause greater damage than what you have foregone, it might be worth it -- but this is almost never the case. Combined with the fact that some magi find it offensive, this makes Pumilius probably the second worst style after plain Gladiator.
Retiarius: OK. Taking a bonus up to your Finesse to Initiative in exchange for taking the same penalty to the first three attack rounds seems a poor choice -- to a very rough approximation, gaining you one attack while giving up three. That's not quite always true: for example, if you are the weaker magus, but can spend much more vis than your opponent (and thus you want to spend it as soon as possible, before he pummels you to unconsciousness). In some cases, winning initiative is paramount, and Retiarius is the only style that improves your chances to achieve that. Still, the circumstances in which winning initiative might be worth the cost are sufficiently rare that Retiarius is not worth more than an "OK" rating.
Scissor: GOOD. Scissor is another style that looks worse than it really is. Scissor allows you to forego attacks in a round, in exchange for achieving more damage on a successful attack on the next (Weakening-Resistance Total gets divided by 3 instead of 5 to determine Fatigue lost). Looking at it naively, it seems that you are getting somewhat less than twice the return on attacks, in exchange for attacking no more than half as often -- a clearly losing proposition against even plain Gladiator. The catch is that often you can forgo weak attacks and "boost" strong ones. One example is when using vis to attack: scissor allows you to maximize its effects. Another is when bringing into play highly unbalanced Arts: in these cases, you can choose your high Art for defense in those rounds in which you have to forgo your attack, and for attack in those rounds in which you can take advantage of Scissor's bonus. Interestingly, Scissor is one of the very few styles with dedicated practitioners (and the only strong one) in which Finesse does not enter any total (save the basic Initiative total), so it's an excellent style for young magi and for magi whose Finesse is somehow impaired.
Velitus: GOOD, verging on GREAT. Velitus is to Laquerius as Provocator is to Hoplomachus -- affecting Resistance instead of Defense. The other fundamental difference between Laquerius and Velitus is that Velitus can inflict (in exchange for foregoing damage) half the attacker's Finesse as a penalty to the defender in all future rounds; whereas Laquerius can inflict the up to the whole Finesse -- but at the price of taking an identical penalty to the attacker's own defense in the round of the attack (before knowing whether it will succeed). In some sense then Velitus is a "tamer" version of Laquerius, giving a lesser bonus for a significantly lesser penalty. So for most magi Velitus would be a better choice. But for that majority of magi, neither Laquerius nor Velitus is the best choice: Velitus and Laquerius both shine only for a minority of magi, but, generally speaking, for that minority Laquerius tends to shine a little brighter, because it allows you to build an "unstoppable" advantage over the opponent more quickly.