Shukhov definitely doesn't own much, but the few things he does own deserve special mention. We hear a lot about Shukhov's spoon, which acts as a little memorial to his time as a prisoner since Shukhov carved when and where he made it into the handle. In a way the spoon serves the same function as Shukhov's practice of removing his hat before meals: both gestures are a way for Shukhov to hold on to some sense of self and his past in the camps, instead of just existing on a day to day basis as a mindless robot. Shukhov works hard not to lose himself or his past in the camps.
Shukhov's other notable possessions also help to give us increased insight into his character. He owns a needle and thread, and he plans to make a bread knife out of the piece of steel he finds. He also has a trowel that he hides for safekeeping at the worksite. All of these objects are signs of how industrious and hard-working Shukhov is, as well as evidence of how well he plans ahead. He takes good care of his possessions, and he uses them to help him survive.
Finally, we have the lost possession that Shukhov takes the time to mourn: his lost shoes.
He'd taken such good care of his nice new shoes, he'd greased them to make them soft [...] He'd never missed anything so much in all those eight years. The shoes were all tossed on one big pile [....] (58)
Shukhov's shoes represent how cruel camp life can be – everything is stripped away from inmates in the prison, and prisoners can lose their belongings at any time. Since Shukhov has almost nothing that is his own, his belongings become hugely important to him, and are almost like friends. So losing something like his much-loved shoes is really quite painful.
This may also be a somewhat subtle dig against the sort of communism practiced in the Soviet Union. Private property was "bad" and the government went around seizing people's belongings, much as Shukhov's beloved shoes are tossed into an impersonal pile.