Spelunky HTML5 Saving
Solution 1:
You can't save your progress in individual runs in SpelunkyHD (or the executable version of the original Spelunky), but it does keep track of your statistics (deaths, high scores, enemies/places/items discovered) and contains unlock able characters not available in the free version of Spelunky.
This question will outline the differences between HD and the original (the HTML5 version is a clone of the original version) further:
What is the difference between Spelunky and Spelunky HD?