train bookings in advance in May?

I'm planning a trans-Siberian trail trip in early/mid May and need some advice on train bookings. I'll be arriving in Moscow, and will spend a couple days there before taking the train to Irkutsk, and onward to Ulan Ude and Vladivostok.
I've heard conflicting reports on the need to book rail tickets in advance- some say it's absolutely necessary, others say there's always a train. Am I safe to book my rail tickets upon arrival in Moscow, or will I need to book in advance through an agency? I'm looking to take either the #2 or #10 from Moscow.

For trans-siberian train I would make the reservations in advance. Generally speaking local trains have always been no issue for myself, I have experienced some longer distance trains selling out - both times I was able to get another train in a few hours. Unless your travelling with someone thats Russian I'd say why chance it.

I'm not the expert at train travel in Russia though, so others advice might be better.

Regards