In the NHL, there have been 41 games tied 0-0 after overtime since the introduction of the shootout in 2005 (meaning they would have been recorded as a 0-0 tie in the past), which is 0.25% of games. And there have been 189 0-0 ties in history (dating back to 1924) [games are double-counted in link]. There have been 122 games that were 0-0 after regulation only to be ended in overtime (not a shootout). This gives 352 total 0-0 games after regulation. Note that this is just regular season, and not playoffs.
In the NFL, there have only been two 0-0 games [games are double-counted in link]. The last one occurred in 1943. I couldn't find any other games that went into overtime tied 0-0 (checked each box score individually), either, which surprises me. Those 0-0 ties were Detroit against the Chicago Cardinals, and Detroit again against the NY Giants.
For the MLB, I was able to find at least 620 games that were tied 0-0 after 9 innings by looking up games where that took extra innings and the losing team had 0 runs. There appear to be a handful every year by changing the dates. I am not a subscriber, so I cannot verify the total or see more games. This also does not take into account games that were tied 0-0 after 9 innings where the losing team managed at least 1 runs, so there are more.
In the top 4 tiers of English soccer, from 1888 through 2014, 13,475 games (7.2% of matches) ended 0-0.
Seems like it's nearly impossible in the NFL, not too uncommon in the NHL, very common in the MLB, and almost weekly in English soccer.