Which parts of the ocean have been explored?
The simplest way to find out the basic information about your nearby ocean is to check Google Earth, which has an Ocean layer that often includes details about noteworthy features and discoveries.

This story from Scripps Institution of Oceanography explains some of the updates that were made in late 2015:


  The new global gravity map, recently published by [geophysicist David] Sandwell and
  colleagues in the journal Science, was used in combination with
  available depth soundings to provide a more detailed picture of the
  mountains and valleys along Earth’s most unexplored expanse – the
  ocean floor.


This essay in Scientific American helps explain some of the extent of our knowledge:


  So the “95% unexplored” meme doesn’t really tell the full story of our
  exploration of the oceans. When it comes to having a large-scale map,
  the ocean floor is perhaps not as unexplored as we might think, with
  100% coverage to a resolution of 5km and 10%-15% coverage at around
  100m resolution. That 10%-15% is similar in resolution to the current
  global maps of Mars and Venus.
  
  But our exploration of the oceans depends on what we want to know
  about them. If our questions are: “What does it look like down there?”
  or: “What’s going on down there?”, then the area that has been
  “explored” is arguably even less than the 0.05% mapped so far at the
  very highest resolution by sonar.


The areas mapped in most detail will be near coasts and common ship routes. Oceanographic vessels run their sonar systems almost continuously, so the maps will have lines of very high resolution detail running through them (artifacts that are often interpreted to look like lost cities or roadways).

But surprising discoveries are still made. It was only in 2012 that a Scripps expedition found a methane seep that was later determined to support an elaborate chemosynthetic ecosystem just 20 miles off the coast of Del Mar in Southern California.