I don't remember any truly spurious accusations against him.
The only one I can think of is the MLK story.
In 2009, Obama removed a bust of Winston Churchill (lent by the UK to George W Bush after the 9/11 attack) from the Oval Office, returning it to the British Embassy in Washington [1].
Obama had a bust of Martin Luther King on display in the Oval Office.
After Trump's inauguration, a WH press-pool reporter noticed and reported on Twitter that the Churchill bust was back in the office (it was), then added that the MLK bust had been removed (it hadn't, it was simply obscured by a door).
It was never really a story - just a handful of tweets - and the reporter tweeted the correction within an hour [2]. Trump has used this as an example of "media dishonesty:".
[1] http://www.telegraph.co.uk/news/worldnews/barackobama/462314...
[2] http://www.snopes.com/mlk-bust-oval-office/