Where was Sirius's bike all the years?

Question

Please read my update below

This is just a simple question, that came to my mind while I was listening to the Deathly Hallows and heard:

Hagrid was standing ready beside the motorbike, goggles on.

“Is this it? Is this Sirius’s bike?”

“The very same,” said Hagrid, beaming down at Harry. “An’ the last time yeh was on it, Harry, I could fit yeh in one hand!”

The bike was introduced in the very first chapter of the first book:

‘Hagrid,’ said Dumbledore, sounding relieved. ‘At last. And where did you get that motorbike?’

‘Borrowed it, Professor Dumbledore, sir,’ said the giant, climbing carefully off the motorbike as he spoke. ‘Young Sirius Black lent it me. I’ve got him, sir.’

I assume the motorbike was not retrieved back to Sirius, since it was about the time when he was accused of killing thirteen muggles and sentenced to the Azkaban. At least I don't know anything from the books which would hint at any other conclusion.

I assume Hagrid had the motorbike. But, where was it all those years? I mean, Hagrid claims after Dumbledore's death:

“Well, I’m stayin,” said Hagrid, fat tears still leaking out of the corners of his eyes and trickling down into his tangled beard. “It’s me home, it’s bin me home since I was thirteen. An’ if there’s kids who wan’ me ter teach ‘em, I’ll do it. But… I dunno… Hogwarts without Dumbledore…” He gulped and disappeared behind his handkerchief once more, and there was silence.

So it looks like he was almost all the time there (and I think so was Sirius's bike). The motorbike might be some time at the Burrow, but I would say for only a few weeks before the transport:

“Arthur’s done a bit o’ tinkerin’,” said Hagrid, quite oblivious to Harry’s discomfort. He settled himself astride the motorcycle, which creaked slightly and sank inches into the ground. “It’s got a few tricks up its hindquarters now. Tha’ one was my idea.”

I do not remember the bike being mentioned anywhere else - even, when Sirius was at Grimmauld place. Harry is definitely suprised, like he never saw that motorbike before - and there is no reason not to believe him. I can't think of any other place, where the motorbike might be hidden other then Hagrid's hut.

I mean, this scene definitely adds the good nostalgia to the scene, remembering the very beginning, but I can't help myself thinking, that the motorbike disappeared after that.

Question. Any ideas where the motorbike might have been?

Update 2^nd June

I did not want to accept @Anthony Grist's answer because I still thought there was something in the books. And I think, there is. Please take a time to read:

“No, Harry, dear, I’d much rather you helped Arthur much out the chickens, and Hermione, I’d be ever so grateful if you’d change the sheets for Monsieur and Madame Delacour; you know they’re arriving at eleven tomorrow morning.”

But as it turned out, there was very little to do for the chickens. “There’s no need to, er, mention it to Molly,” Mr. Weasley told Harry, blocking his access to the coop, “but, er, Ted Tonks sent me most of what was left of Sirius’s bike and, er, I’m hiding – that’s to say, keeping – it in here. Fantastic stuff: There’s an exhaust gaskin, as I believe it’s called, the most magnificent battery, and it’ll be a great opportunity to find out how brakes work. I’m going to try and put it all back together again when Molly’s not – I mean, when I’ve got time.”

_{Harry Potter and the Deathly Hallows, ch. 6 The Ghoul in Pajamas, pages 84-85}

Seems to me, that for one thing, he is hiding it in the coop, and for second, it really does not sound, like he had the opportunity for the last sixteen years to play with. One might say, that it's because it is now crashed and demolished, but I think Mr. Weasley he likes to find out more about Muggle inventions whenever he can:

Mr. Weasley liked Harry to sit next to him at the dinner table so that he could bombard him with questions about life with Muggles, asking him to explain how things like plugs and the postal service worked.

“Fascinating!” he would say as Harry talked him through using a telephone. “Ingenious, really, how many ways Muggles have found of getting along without magic.”

_{Harry Potter and the Chamber of Secrets, ch. 4 At Flourish and Blotts, page 37}

So my assumption from this is, that he could not have had the bike all the years.

Answer 1

There's not a whole lot else from the books that's relevant, and this is all I can find from JKR herself on the matter:

Rita: Whatever happend to Sirius' flying motorbike?

JK Rowling replies -> Ah, good question. You'll find out, but the real sleuths among you might be able to guess.
_{World Book Day 2004 on Accio Quote!}

With that in mind, I think you have the answer in your question already:

“Arthur’s done a bit o’ tinkerin’,” said Hagrid, quite oblivious to Harry’s discomfort. He settled himself astride the motorcycle, which creaked slightly and sank inches into the ground. “It’s got a few tricks up its hindquarters now. Tha’ one was my idea.”

Arthur Weasley had it. There's absolutely no room in Hagrid's hut to hide it, since it is precisely that: a hut. Arthur having it also makes sense, since he also had the Ford Anglia that he enchanted to fly, and he's absolutely fascinated with Muggle technology. It may well have been the inspiration for the enchantments on the car in the first place.