These headphones are tuned exceptionally well. Unfortunately, I have not heard the AKG K371 so I cannot compare them to their sibling cans, but from what I am told, the K371s have a bit more excitement in the bass at the cost of less soundstage.
The K361 is really well-tuned, with accurate tonal balance throughout the whole frequency response. These headphones are also very comfortable, with an all-plastic build that makes for a very lightweight and fatigue-free fit. It's mostly soft-touch plastic that feels rather sturdy in the hand. I imagine most folks could wear these things for hours without any particular problems with comfort.
On The Alan Parsons Project's "Don't Answer Me," the soft background vocals in harmony with the chorus can often fade into the mix on poor headphones. This is especially the case around 2:14 during the lyrics "Run away and hide from everyone." Fortunately, the AKG K361s present the background vocals just fine, separating the vocals well enough to differentiate the background vocalist in the left channel from the lead vocalist in the center and another background vocalist in the mid-right area. It's that mid-right background singer that can often gets buried, but I can pick him out plenty well on the AKG K361.
Transients are also smooth, these drivers certainly run quick enough to make sense of them. On the track "Bremen" by PigPen Theatre Co., vocals have a very gentle reverb on them that can often get buried by the guitar and banjo. The fast decay is audible with these cans, making it easy to track exactly when each note finishes.
Sub bass definitely seems favored over mid bass, with a fairly clear low-end- an exciting but well-balanced bass. I like this pair of headphones for the tune "あめあがりのうた" by Snail's House. I don't consider myself much of a bass-head as I certainly prefer bass quality over bass quantity, and this is one of those tracks than can give me a headache when I play it on bass-heavy headphones. Fortunately, the low-end is present and with plenty of excitement, but tame enough for my ears to really enioy on this track.
As the title suggests, this pair of headphones has one fatal flaw. The biggest problem with the AKG K361s for me would be that it's nigh impossible to get a good seal while wearing glasses. Seriously, the first time I put thes headphones on, I thought something was wrong, they didn't sound right. Then I took my glasses off and they sounded amazing. Looking online, it seems Dekoni Audio makes some pads for the AKG K361/K371 that are alleged to work a lot better for getting a good seal with glasses, but that is adding $50 to a headphone that costs only about twice that. In total one would end up having to pay about $150 total for the headphones and their remedy. I do believe that's still a fair price for what you get, but at that price point one might consider a different pair of headphones instead.
In case you're curious, the headphones come with a fair assortment of goodies, including two cables, one of a short length and one that is longer, as well as a soft tie-up case and a screw-on quarter inch adapter. It's nothing particularly significant, but it's also nice to have the different options for cable length. The cables plug into the headphones with a 2.5mm TRRS end that twists to lock into the left ear. Both cables terminate in 3.5mm single-ended that can be used with the screw-on quarter inch adapter.
The headphones also fold up for compact storage.
I've had a pair of the K240s for a while and I have to be honest and say they are my least favorite headphones in the collection. The soundstage is good, but they are totally bass-anemic. Vocals are clear on the AKG K240 but I feel like that's all they really have going for them.
I don't know which particular version of the K240 you have, I'm talking about the newer "AKG K240 Studio" that are rated at 55 ohms of impedance. It's possible that older sets rated to be less sensitive have better tonal balance, but I feel these Studio ones are simply not it.
I will say, however, I used the K240s when I was hosting a radio show and they were phenomenal for that application. Being able to talk into a microphone while also monitoring levels and not getting that awful feeling of speaking while wearing closed backs was great.
At this point, I feel there are simply better, more balanced options that can offer even better soundstage with tonality that excels magnitudes beyond the K240. Sennheiser HD 6XX and Hifiman HE400SE most immediately come to mind.