Where are the locations of the Lost Sectors?

Solution 1:

Lost sectors are small caves that are found all over destinations. They are marked in the world by a symbol of a dot with 2 arches above it to indicate the somewhat hidden entrance nearby. They are completed by killing a yellow-health barred (sometimes named) boss and acquiring a cache code and then opening the special chest at the end. There are lots of other enemies that will get in your way but they can be killed or ignored if you wish. Once completed the first time, the symbol on the map will become grayer but still noticeable for you to find again and redo for more loot or to complete a daily challenge.

Here is what the symbol looks like in the world near an entrance.

Here you can see the same symbols around the map at many locations. Note the white (uncompleted) and gray (completed) symbols in the Trostland.