With the introduction of transmogrification in patch 4.3, collecting old armor styles has become one of the most productive ways to spend your free time in World of Warcraft. As an added bonus, soloing old content (or going in very small groups) can be very lucrative. I made 2,400 gold tonight from soloing Karazhan, Gruul's Lair, and Magtheridon's Lair and grouping for Sunwell Plateau, Tempest Keep, Serpentshrine Cavern, and Caverns of Time: Mount Hyjal.
Both specs of DPS shaman bring some excellent soloing utility to the table. As mail wearers, we have some solid physical damage mitigation. Wind Shear is an incredible utility ability that will be used on bosses in nearly every tier of raiding. Earth and Fire Elemental Totems provide a second target and work especially well for pulling council fights, so that you can pick off one target and kill it while the rest attack your elemental. Tremor Totem is useful on fights like Lady Vashj. Grounding Totem can be used to absorb direct damage spells your interrupt is down for or catch boss CCs that are headed your way. Bloodlust is great for burst DPS, and especially helpful on fights that have a soft enrage. And, of course, being a hybrid class means that both of our DPS specs have some amount of self healing to bring to the table.
The easiest way to solo older content is by speccing enhancement, provided you have some solid level 85 agility gear to support the spec. Enhancement brings a lot of benefits to the table. It has some great healing in Maelstrom Weapon five Greater Healing Waves, which can be backed up by the passive healing of your Feral Spirits when they're up. Shamanistic Rage is your own private Shield Wall for when the going gets tough. Dual Wield gives us the ability to parry, which is a small bonus when fighting melee-heavy mobs.
Enhancement has its downsides. The low mana pool means that tight spots can be a struggle to escape from. Three non-Maelstrom Greater Healing Waves will annihilate your mana pool. Moreover, you lack the ability to attack from a distance. On single-target bosses like Gruul, this is immaterial. However, on bosses like Nightbane (who has an incredibly long air phase at every 25% of his health) or council fights like High King Maulgar (where you need to outrange Death Coils and Polymorphs), the disadvantages of being a melee character are clear.
If you do want to go enhancement, there are a few changes you'll want to make for your soloing spec. The top DPS enhancement spec puts two points into Improved Shields for some added Lightning Shield damage. That's not really needed for soloing; grab those points and an extra one from elemental to max out Toughness. I also suggest grabbing Spark of Life to increase your self-healing. You also want to grab Focused Insight; since shocks are a pivotal part of your priority, the Focused Insight proc will almost always be up, making all of your instant Greater Healing Waves hit that much harder. Your last talent point is a floater. Acuity will help damage and healing, while Nature's Guardian can rescue you from a sticky situation. I go with this spec for enhancement soloing.
That said, there are benefits to going elemental over enhancement. When specced properly for soloing, elemental has tons more passive damage mitigation than enhancement. First, elemental has the benefit of using a shield, which provides around 15% more physical damage mitigation than enhancement. Add Elemental Warding on top of that, and an elemental shaman passively takes 15% less physical damage and 10% less magic damage than enhancement. Add in Ancestral Resolve, and your elemental shaman becomes even more tankish, taking 10% less damage 90% of the time. Elemental also has more solid burst damage than enhancement, due to Elemental Mastery.
Just like enhancement, elemental has its own downsides. The most obvious one is spell pushback, since you'll be taking melee swings quite often. This can be somewhat helped by Totem of Tranquil Mind, but using that will cause you to lose out on Healing Spring Totem and your resistance aura, which is painful. Ele can cast more heals than enhancement can but doesn't have the passive mana regeneration enhancement has in Primal Wisdom. It also lacks any sort of damage reduction cooldown unless you glyph Elemental Mastery, which makes escaping from tough situations harder.
Also, you'll want to subspec restoration rather than enhancement. Ancestral Resolve and Spark of Life are too incredibly powerful to pass up. You have the choice of going for Focused Insight or not, but with no Fulmination you'll only be shocking every 27 seconds or so. You can also try putting points into Elemental Weapons in enhancement for some added DPS. I personally put the rest of my points into Tidal Focus and Nature's Guardian, giving me more survivability.
Glyph of Unleashed Lightning will be your best friend soloing. Get used to strafing away from bosses while casting, as strafing will allow you to cover more distance. You still want to go with Glyph of Flame Shock and Glyph of Lightning Bolt (though going Glyph of Lava Burst would not be a huge difference). For majors, Healing Spring Totem and Stoneclaw Totem are both vital. Drop Glyph of Lightning Shield. You're no longer using Rolling Thunder or Fulmination, so there's no point in even using Lightning Shield while DPSing. Instead, go for Water Shield, as that will give you more mana for healing. Your last spot has two main choices: Glyph of Thunder is more mana per minute, and Thunderstorm does some nice damage itself. Glyph of Elemental Mastery gives you a defensive cooldown (which elemental sorely lacks), but it is tied to your offensive cooldown.
No matter which spec you choose, soloing at least the first tier of Burning Crusade content is within your reach. I've not had the chance to test either spec in later BC content myself, but I'd love to hear from other players if they have.
Show your totemic mastery by reading Totem Talk: Enhancement every week. We've got enhancement-specific advice on the latest gear and tier gear set bonuses, and we'll help you dig in and learn to level the enhancement way.