Interesting topic! Come to think of it, I somehow most of the time skip most of the shrines, even when I am playing single player pass. The ones I always use are (mind you: I strictly speak about single pass untwinked tournaments)
1. Gem Shrine (in early stages to upgrade Topazes for more MF in single pass untwinked play, later to upgrade a flawless game I need most of, e.g.for certain crafts)
2. XP Shrine. I usually try to find at least one boss pack before activating it or I activate it shortly before entering an area with lots of boss packs expected.
Then there are the shrines that depend on circumstance:
3. Resist Shrines. I sometimes save them for dangerous boss packs or act bosses, esp. If they are near a WP. Getting to 85% poison resist before facing Andariel is useful from time to time
4. Stamina shrines: Esp. in early game I hit them regularly ro avoid rests. Saves a few seconds.
5. Weapon shrines: very seldomly used for dangerous fights or tedious boss fights if I rely on weapon damage (e.g. I would have loved such a shrine for various boss fights with my random amazon - 30% cth with ~200 damage in hell makes some fights against fast regenerating bosses very long)
Skill shrines and Armor shrines I somehow always never activate, although it for sure is worse to not click them than to click them... After thinking about it, it is probably because I fear I might need the shrine later but then never use it
Explosive and Poison shrines I usually always click on early stages and even pick up the potions one by one (one for each slot) to sell and rebjy them individually if necessary. They are quite strong in the early game, esp. for tournaments with rough restrictions. And even if I don't need them, they are worth dearly needed gold.
Fire Shrines I sometimes use against bosses with lots of hit points, but in HC that may be a two-edged sword...
I don't know if I forgot any (Wells and health/mana shrines I use as needed, but I always try not to rely on them in HC).