In League of Legends, you can attack wards for a short window of time when they're put down by the opposing team (discounting the use of other methods of finding wards).
Often, a quick autoattack at the beginning of the match can only take out one segment of health on a ward, and since I can't find a definite consequence of wards losing health, I was wondering if there's a benefit to attacking wards if you can't take it out entirely at once.
This is a tricky question, as the answer is HIGHLY situational. Wards tend to be used as bait, beyond their uses for vision. Overall, you generally want to remove enemy wards as quickly as you can, without risking yourself or teammates. Denying the enemy vision is extremely important, but so is not dying. Basically, if you can't destroy a ward, attacking it is a good idea, unless it risks your safety. Let's go over some example situations.
You're in lane, and see one of the opponents drop a ward. If your basic attack is ranged, go for a shot on it, or if you happen to be standing right next to it. Again, if there are clear risks, don't go for it. The benefits you are getting in exchange for that one hit, is that it is easier for your support or jungler to clear. That extra second or two that it takes to hit a ward 3 times as opposed to just twice, can be the difference between clearing and not clearing a ward, or clearing it in time to get out safely. This is especially true when the support is the one clearing the ward, as they generally don't ever have an attack speed of more than once per second.
A second situation to consider is when destroying wards is true sight. If your team has true sight in the area, not granted by a pink ward (Although, this situation may apply if you know for a fact they know the pink ward is there). If an enemy places a ward in an area that they KNOW you have true sight, then there is a 99% chance that the ward is a bait of some sort. So what do you do? Simple, steer clear until someone on your team can disable it with a sweeping lens, the kill it. You want to disable the ward FIRST to deny the enemy vision of your team moving to kill it. Attempting to kill a ward they know you can see without disabling it, will likely result in a Blitzcrank Rocket Grab Q, a Lee Sin jumping to you with his Sonic Wave Q into Resonating Strike Q combo, a Nidalee Javelin Toss Q to the face, or some other equally nasty surprise.
Next there's the Inhibitor Tower Siege, Over the Base Wall ward. Do NOT under any circumstances try to destroy these. I guarantee that there is an enemy waiting for you to get close to hit you with something nasty. The best way to deal with this is to counter ward, by placing your own ward over the wall as well. This way, you can watch what they are doing, and you spend much less time in the danger zone than if you are attempting to destroy it. The only exception is again, if an ally disables the ward with a sweeping lens. Be wary though, your opponents are much more likely to attempt a blind skill shot in this situation than in the above. You will, however, be safe from targeted spells such as Talon's Cutthroat E.
The last is that random ward an enemy tossed down as a team fight broke out. Leave it, but memorize it's spot. It's not important enough to stop focusing on the enemy damage dealers to kill the ward as fast as possible, but you do want to be able to have an ally hit it with a sweeping lens once the action calms down.
TL;DR? There are lots of benefits to getting that one free shot off on a ward, but in most situations, the potential risk is not worth the one hit on the ward.
Yes, it is worth hitting the ward. Sometimes if there are 2 of you there, you may manage to actually kill it. You or someone on your team might reveal it later and requiring to hit it less keeps them safer.
As long as it doesn't put you in danger, it is always worth hitting the ward.
You attack wards as a general principle.
Unless attacking the ward will put you in a hard position there's no reason why you shouldn't attack it, really.
Well, there is no point on attacking a ward if you can't destroy it. There is no benefit at all. But there are select champions who can hit wards three times in quick succession. Leona for example, she can do an autoattack-Q-autoattack combo and right timing to destroy a ward just before it goes invisible. This is one of her strong points, denying enemy vision. I think Garen can do this as well.
The only real benefit to attacking a ward that you cannot kill in 1 sitting, is if you are planning on revealing the ward later and killing it. For example, using the new reveal trinket, you can hit it once, and then reveal and only have to hit it twice more. It might not seem like much, but for example, you might be putting yourself in danger by revealing and destroying the ward, and killing it in 2 auto attacks is a lot faster than with 3.
When should I spend my auto-attacks on the enemy wards?
Is it worth to just hit a couple of times even none of the criteria above fits?
If you don't have nothing else to do, go for it. Besides passing time and put yourself at enemy range/risk. NO isn't worth at all.
Worth to mention notes:
TL;DR;
Don't hit it if you are not going to destroy it. Be careful!
