This thing has bothered me a lot. I want to buy house in Riften and find a way to become their Thane but it seems impossible. I have tried to find quests from people near Laila Law-Giver but only I find was that her son needed some help. I helped him but still there was no chance to buy a house. I have also helped several other people (only avoided criminal quests/factions) in Riften to get some name there but no - still nothing. Why? Is that even possible to own house or be a Thane in Riften?

Oh... [censored]. I think there is one related quest you must get and prior to the civil war concerning Riften. That would be dealing with the Skooma dealers... 3 parts I believe. Pretty darn sure that will get you favor... and if not, you should be able to buy one after the civil war is over, and the Jarl is replaced. Not sure about these, but I do remember reading them in the guide. Get one. It's worth it.

It is possible
You've done part of the work with your various good deeds around Riften but theres one particular quest chain you must do
You need to find an Argonian called Wejeeta (or something like that) and help her with her problem to get started. She can usually be found around the docks area since she works at the fishery

You need to find an Argonian woman looking for help with a skooma problem. She works in the fishery but comes in at the gate in the evening, that's usually where I see her.

I just wish you could become Thane of Riften without committing a crime. Well, you don't actually commit a crime, since you're acting as a law enforcement agent, but it counts as a crime in your stats sheet. My "no crime stats" character became Thane everywhere but Riften, even though Riften is my town of choice. Still bought the house due to the Civil War, but still.

What crime to you have to commit to complete Thane of Riften?

If you handle Riften to the imperials, the son of the new Jarl will give you the house without having to perform menial task for that other incompetent Jarl.

Entering the warehouse always counts as trespassing. It's not a big deal and you won't get a bounty, but as I've said, I didn't want any crimes listed in my stats. There didn't seem to be any way around that.

If you start the war there will be counted several murders in your crime stats - even you have committed no murders. I don't know why it is so but it's connected somehow to wiping out enemy camps. So one counted trespassing for getting the title of Thane is nothing.

Nope. I've completed just about everything in the game, with over 70 hours logged, and my crime stats are still zero.

Is it so? I always get one or two murders counted in my stats when I walk in enemy camp and kill everyone who are attacking to me. I have to ask have you ever killed soldiers on enemy camps?

I take it your not going to pick any locks either?

Make sure you read about this on one of the wikis because there's a bug (shocking, I know ) wherein if you go to the house after you buy it, but before getting the title Thane of the Rift (I believe - as opposed to Thane of Riften) you don't actually become Thane. I got the message and dialogue that I was Thane, but when I tried to kill the Chance Arrangement quest by deliberately getting caught picking the lock on Madesi's display case I wasn't offered the "I'm a Thane..." dialogue option. Also, even though I was give 2 weapons by the Jarl I do not have Blade of the Rift.

Yes, it's possible. My character has a house in Riften

I also got the Honeyside house after doing the Jarl's skooma quest line. But like patti 345 says, that questline is bugged. If you make the mistake of visiting your new house before talking to the Jarl again, then you'll break the quest. She'll never bring up the subject of you becoming Thane again. So you'll never get your housecarl. I had to use the console cheat to force an end to the quest and automatically make myself Thane. That was the only way I got Lydia to show up at the house.

wrong house lydia is in whiterun not riften

You also get lots of assaults if you are sneaky, some bandits are not hostile until you come close and if you attack them first it's assault.

The tower standing stone was my best friend. Doesn't count as a picked lock.

Ooooo, I wonder if that's why I have a murder in my Stats. I have one murder and had a 1000 Septim bounty in Falkreath after only visiting the town once - I never knew whence it came. I bet I killed a Bandit (or initiated the attack) from a distance with a bow.

I've bought the house in Riften several times. I've never done either of the Civil War questlines, so I can't get any of the others.