The adept is definitively one of the more powerful and fun classes to use. The main skill they use is singularity which lifts enemies into the air and renders them defenseless for a considerable amount of time.
The adept mainly focuses on the support role, but will often get the most points of the team in multiplayer. This is down to the effect of combining powers such as overload and biotic charge which triggers a biotic explosion. The best way to achieve this is to alternate singularity and throw which will down even an Atlas quickly. However, because the adept doesn’t have a shield recharge skill, you have to be careful with the way you move around the map and be sure to use cover a lot.
The soldier is one of the most boring classes to play as. All of your skills are based around enhancing your use of guns and melee which is not what the developers of mass effect 3 spent their most time on. However, I found that using a Krogan is very fun because it is different from all of the other classes in that he is one of the strongest tanks in the game. Krogans are mainly designed for melee combat which is hard to master. I have written a full guide for using the Krogan effectively here.
The engineer is another example of a support character. However, unlike the adept, I have rarely seen them top score in a multiplayer match or be of any use. The combat drones are somewhat useful distractions especially when you are struggling to kill a Guardian because he has his shield pointed towards you, but most of the other damage has to be dealt with weapons.
Traditionally the engineer is very dependent on teammates to do most of the damage, especially in the single player. It has never been a strong choice in any respect which is why I haven’t decided to test it out very much. With that said, the incinerate is a very useful tool to strip armor away but, especially in multplayer, only the Atlasses have armor which renders that skill useless in most situations.
If played right the engineer can be very powerful. I haven’t seen this very much, but if you specialize overload so that you can spam it very quickly it definitely is very powerful. You can find a detailed guide to the effective use of the engineer class here.
Sentinels are much more aggressive than the engineer and the adept, but still are limited to using the cover system a lot since they lack the ability to recharge their shield quickly. Sentinels have the tools to take care of most enemies effectively. They can use warp to take down the armor of an atlas, overcharge to rip off shields and throw to render enemies harmless for a few seconds. Throw is very powerful because it can be put on a very quick cool down given the right specialization.
Personally I am a big fan of aggressive characters that spend most of their time in the middle of the action. However, they are a very useful addition to any multiplayer team and are a strong choice for the single player.
Now I am getting to my favorite classes. In Mass Effect 2 the infiltrator was easily one of the most powerful and easy to use characters. You could alternate between cover and cloak when you had to exit cover to minimize the damage that you would take and when you aim down the sights with a sniper it slowed down time. In multiplayer the cloak has stayed which makes it a powerful class to begin with, but they have added an ability called energy drain which takes an enemies shield or barrier and gives it to you. It even hives you a slight armor boost when specialized properly.
In your aggressive gameplay you are somewhat limited by your enemies having to have shields or barriers to survive easily, but whenever you get into a sticky situation you can use cloak to get away from it. However, your shield does not recharge whilst cloaked. Be sure to use a power towards the end of the shield, otherwise the cool down for the cloak is started after you re-appear. The cloak is incredibly useful for the multiple objectives in the mutiplayer as it essentially lets you go in and do them at any time as it lasts almost the full duration of the hack (or whatever else you have to do). I have written a full guide for the Salarian infiltrator which you can find here.
Even before Mass Effect 3 the vanguard has been my favorite class. I completed Mass Effect 2 on Insanity using the Vanguard and it was a blast, ignoring the times when Shepard says “cannot target” which in most cases was a death sentence. In mass effect 3 the vanguards are powerful to the point of being over powered. You can put charge on a 2 second cooldown and have that even reduced when using nova. You have invincibility frames whenever you are in the charge animation or the nova animation and about half a second after you are out of nova the charge will be ready again.
The vanguard suffers from connection though, I have been downed many times because my character wouldn’t charge because of high ping/bad connection. However, the vanguard will always be where there is clusters of enemies and make it look easy at the same time. You can find a full guide for the human vanguard here.