Jungle Shen is a lot better than it would seem. His initial clear speed is not terrible and the amount of utility he offers makes him a force to be reckoned with. He is amazing at dueling and he can 1v1 most junglers. His ult, while it has a long cooldown, can save an ally and win a fight that you would normally lose. It also gives a huge global presence, allowing you to split push and immediately teleport to allies if the enemy team tries to collapse on you or engage them. He has the added bonus of being able to recover from a bad start, since he can scale hard late game.

Pros:

Tanky jungler
Kit is great for dueling
Can shut down a

Xin Zhao
Provides great utility for team

Cons:

Ultimate has a long cooldown
Difficult to master completely
Very vulnerable to ranged champs
First clear is slower than the meta junglers

This rune page may seem weird but it has a purpose. Normally Shen's clear is bad, but this rune page gives more attack speed, so it makes him viable in the early game, where he would normally struggle as a jungler. Press the Attack gives increased damage to champions and is really the only good precision keystone on Shen, Triumph gives you sustain in team fights, Legend: Alacrity gives you the increased attack speed over time, and Coup de Grace tends to be the best of the final three minor runes. You can also go Last Stand, but never go Cut Down, since you are a tank.

For the runes, there are a few options for Shen jungle. The most common primary tree to use would be Resolve with Aftershock keystone. This is what support Shen uses and it is great for that because it boosts Shen's tankiness. The downside for the jungle is that it takes away your clear speed and the burst damage that makes Shen jungle such a surprising pick. You get nowhere near the attack damage that you have in the Precision tree, even if you take it as a secondary. Domination is almost useless as a primary tree for Shen. As for another secondary tree, this is more forgiving since it will offer more initial damage. It is still not ideal but you could use Domination or Resolve secondary. For Resolve, run Conditioning and Revitalize. This will increase your resistances and boost the shield in your ult and passive. For Domination, run Sudden Impact and Ravenous Hunter or Ultimate Hunter. This will give you a tiny bit of damage and lethality burst and give you either sustain or a reduced CD on your ultimate. The reason I would still choose Inspiration over either of these options is the shear gold value. This build is rather expensive and the free boots and CDR the Inspiration secondary offers is helpful in this aspect.

Smite: No explanation needed. You are a jungler.

Flash: The only other summoner spell I will ever use on Shen. It is good for escapes should you need it and is integral to the famous Taunt-Flash combo.

Teleport: If you really want to, this can be used instead of flash. It is situational, but you can use it as an early game ult kind of thing, or you can split push, ult with your team to win a fight, and then teleport back to keep pushing.

There are really no other summoner spells that are good options for a jungle Shen.

Your first clear is generally just a quick clear to open up a gank. I start with the bot jungle to get a leash from my bot-side. Then I go to wolves, the scuttle on that side, and then finish with the other buff. Then I look to get a gank off for either top or mid lane. If you successfully get a kill on the top lane, camp there and try to get another kill when the laner TP's back. Usually they will be aggressive off a Teleport because they will have a health advantage on your top laner. Do not try to solo objectives like dragon or rift. You do have the health and sustain to do it, but you will take it much longer than other junglers. Chances are the enemy jungler will come by in that time and either steal it from you, collapse on you with his team, or both. It is a really awkward situation. It is best to wait until you gank a lane and get a kill so that you can use your laners to help while the enemy is respawning. Only around level 11 are you tanky enough to solo dragon, since you will be able to tank damage from both the dragon and your enemy jungler should the need arise.

Note: When you are clearing the jungle, note that your passive gives you a shield. To maximize your efficiency, do not activate your W until your passive shield has worn off. This will allow you to block as much damage as possible from camps and keep your clear's healthy. This is especially important against counter-jungle champs.

There are a few combos that will make you a good Shen player. These are universal, whether you are playing Shen Top, Jungle, or Support.

Empowered Q: This isn't so much a combo as an important thing to remember. Your Q Twilight Assault will give you three auto attacks after it that will cause serious damage to enemies. However, if your blade travels through an enemy champion when arriving to you, there will be other effects that will activate. For one, the three attacks will do more damage and have 50% increased attack speed. The target will also be slowed by 35% when moving away from you.

Flash-E: This is a combo to beat a squishy or weak champ who thinks they are out of reach. Basically, you begin Shadow Dash and use your Flash just before it ends. The taunt will carry through the flash and will taunt the target. It will extend the range of your dash and gives a great engage tool. Just be sure that you don't overshoot your target.

E-Q-W: This is your main combo. Press your Shadow Dash and Twilight Assault at almost the same time to dash through an enemy and almost immediately get the three empowered attacks. Press your Spirit's Refuge when your blade arrives at you so that you do not take damage from their auto attacks and watch your target cry.

R-E-Q-W: This is the same combo as above, but it is led off by teleporting to an ally that is in distress. If you have an ally that you can save, activate Stand United on them to grant them a shield. When you arrive, immediately taunt the attacking enemy so that your ally can get some breathing room and do the above combo. Usually your ally will help you and secure the kill. If they do not and you cannot finish the enemy, you can flash or taunt away. For even more cheeky plays, have your ally enter a bush before you activate Stand United. The enemy won't see you coming until it's too late.

While it can be difficult to master, Shen jungle can be a real asset to your team. He has significant damage output and can be tanky to block most of the damage for your team. The majority of his learning curve is the map awareness. You need to always be watching the minimap when you have your ult so that you can know when to save your allies. You also need to be able to judge what allies can be saved, as you will set your ult on cooldown if your teleport target dies. If you have competent teammates, you can also do some serious baits on the enemy. All in all, he is a fun jungler and your enemy will be embarrassed when you walk all over them.
As with all things, League is constantly changing. If you think that you have a better suggestion or have found that different runes or items work better, let me know. I am always looking to improve the guide.

6/11/18 - Change to core build to include trinity.
11/5/18 - re-updated guide. Shen jungle is much more difficult to do now, but it is still viable.
12/13/18 - further updated guide for Pre-season 9. Shen jungle is more difficult. I personally can only play it against a Xin Zhao. Hope you guys can make it work better.