Commands like:
/execute @p ~ ~ ~ tp ~ ~1 ~
don't work correctly (teleporting the nearest player one block up), but instead give out two error messages:
Player 'commands.generic.player.unspecified' cannot be found
and
`Failed to execute 'tp ~ ~1 ~' as <Player name>
where <Player name> is the @p player.
There are three bugs here, but if one gets fixed, the others become irrelevant:

• The command doesn't work like it would if it got executed from the player (which execute should simulate).
• The player name doesn't get put into the first error message. If the target selection for example meets an item, it gets displayed the same way as in the /kill feedback, but not for the player.
• There are two error messages for one error.

If the first one gets fixed, the other two get irrelevant.
Other examples are:

• /execute @p ~ ~ ~ scoreboard teams join test
• /execute @p ~ ~ ~ achievement give achievement.openInventory
• /execute @p ~ ~ ~ clear
• /execute @p ~ ~ ~ gamemode 1
• /execute @p ~ ~ ~ kill
• /execute @p ~ ~ ~ setworldspawn
• /execute @p ~ ~ ~ spawnpoint
• /execute @p ~ ~ ~ xp 1

Special cases:

• /execute @e ~ ~ ~ help shows the help to the player who executed the /execute command once per entity, not to the selected entity itself.
• /execute @e ~ ~ ~ seed shows the seed to the player who executed the /execute command once per entity, not the one selected entity itself.
• /execute @p ~ ~ ~ msg @p test shows <Player name> whispers to you: test and You whisper to <Player name>: test.
  /msg @p test shows the error message Player 'commands.message.sameTarget' cannot be found. The same error message appears for every other entity with its UUID as player description.
• execute @p ~ ~ ~ publish works correctly.