Enviro gave a good starting point with his article on patterns: http://www.goldtoken.com/games/nuggets?id=4

It's always important to avoid patterns in your own play and to catch those of your opponents. In addition, there are a few things to consider:

1) Beat your previous throw. If you happen to win a hand, many opponents feel a sense of frustration and have a tendency to throw the move you just played back at you. I don't know the specific psychology here, but if you are player against a new or relatively inexperienced player, it is often a safe bet to throw the move that would beat your previous throw.

2) Inexperienced players tend to start with Rock. If you suspect you are playing a newcomer, throw paper first. Experienced players tend to KNOW that inexperienced players start with rock and they tend to start with paper. Against an experienced player, a better bet is to start with scissors.

3) If you have an opponent that NEVER plays the same move twice in a row, you can always beat or tie them. If they throw Rock on one turn, you know they will follow with Paper OR Scissors the next, so you throw Scissors and you will tie or beat them. If Paper, then next expect Rock or Scissors and throw Rock. If Scissors, expect Rock or Paper, and throw Paper.

4) Opponents that DO throw doubles generally have a higher chance of throwing something other than their doubled move on the third throw (Unless they have a tendency to pattern with 3-in-a-row throws, of course), and you can use the advice in point 3 to meet or beat them on the following turn.

These tips are geared towards 2 player games, but an applied gamer could use them to predict a field of moves in multiplayer games, too. There are a few more tips out there as well, but we can't give away every trick in one post!