Which classes and weapons are best for countering Combat Mini-Sentries and how?

The following weapons will take out a Mini-Sentry in one (direct) hit:

• Direct Hit
• Homewrecker
• Grenade Launcher*
• Loch-n-Load
• Stickybomb Launcher
• Scottish Resistance
• Any sniper rifle (half charge or greater)
• Huntsman (5/7 charge or greater)
  _{* The TF2 wiki gives this a base damage that ranges below the 100 of a Mini-Sentry's HP, so assuming that's legit it may not work all the time.}

If you have one of these weapons, it should be no trouble to pop out from behind cover and peg the Mini-Sentry before it does too much damage to you. Shooting them from out of range is better, since buildings take consistent damage regardless of distance, so if that's possible it is preferred.

Aside from those weapons, other Rocket Launchers aside from the Cow Mangler will deal 90 damage on a direct hit, leaving the Mini-Sentry with 10 HP so any splash should finish it. Shotguns, scatterguns, and miniguns can deal with Mini-Sentries pretty fast at close range, but it may be difficult to get into such range due to the knockback, and the small target makes bullet spread very significant outside of point-blank (Scouts that beat the knockback might die anyway if too many bullets miss). While inefficient, spraying pistol fire (or anything, really) in the Mini-Sentry's direction from out of its range will eventually wear it down (as it cannot be repaired, any damage is good damage). Sapping them is of course effective.

In addition to Toomai's answer, the Red-Tape Recorder works wonders against Mini-sentries.

This works because Normal Sappers take about 3-4 seconds to kill a minisentry, because it slowly "saps" health until the sentry dies. With the Red-Tape, it merely "reverses" the construction of the sentry, which takes 10 seconds. Note that most engineers will rarely ever knock sappers off of sentries; the cost far exceeds the cost of rebuilding a mini-sentry.

That means you as a spy, and the rest of your team gets 10 seconds of no mini-sentries, and the engineer has to wait out the debuilding process in order to rebuild another sentry.