in Klondike-Solitaire, how likely is a deal with no legal moves?
In Klondike Solitaire there are 7 cards face up on the table, 21 face down on the table, leaving (out of the 52 card pack) 24 cards in the deck. If you're dealing 3 cards at a time, only 24/3 = 8 of these cards are available. So only 15 cards are available at the start of the game.

In order for there not to be any valid moves at the start, you would need:


No aces
No adjacently numbered cards


This is easily achievable with 15 cards. For example:


4 Kings
4 Jacks
4 Nines
3 Sevens


According to Wikipedia, which answers the probabilities:


  
  About 79% of the games are theoretically winnable
  The number of games a player can probabilistically expect to win is at least 43%
  In addition, some games are "unplayable" in which no cards can be moved to the foundations even at the start of the game; these occur in only 0.025% of hands dealt
  


You are therefore far more likely to win the game (even if your decisions are far from perfect) than get an unplayable/"null" hand.