The following is the location and requirements for all of the agents I have found thus far in the game sorted by location. Some of the agents will require either a certain party member with you, or the appropriate knowledge perk from the inquisition perks. Some maybe recruited from knowledge that I didn't have at the time I recruited them. Some of these may also be recruited with different party members, I have only listed the ones I had with me when I recruited them. There are at least two new agents that have been added in the Jaws of Hakkon DLC and they do count towards this achievement.
Thank you punky0827 for that information and to SilentlyForgot for the names of the new agents.
The Hinterlands:
Lord Berand and Speaker Anais can both be found in Lornans's exile. Lord Berand will join just with talking to him. He is on the second floor in the pub.
As for Speaker Anais you have to first clear the rift in the cave in the back of the area. After that you can return to Speaker Anais and she will give you 3 conversation choices. If you chose to have them help the refugees she will NOT join you as an agent. But if you chose to either have her become a spy for information, or spread word of the Inquisition then she will join as an agent.
Witty Ritts is gotten during the quest "Strange Bedfellows". You will have to save her from some templars. After you save her you can find two hidden items nearby that will identify an apostate mage and I can not remember what the second is but it is in the same location. Once you have done that you can speak with her. There are two ways you can recruit her. The first is to have Varric talk to her via the special option in the dialogue wheel. The other is if you have the criminal knowledge inquisition perk.
Horsemaster Dennet is recruited during the quest “Horses For The Inquisition" when you talk to him you can either recruit him via having Cassandra talk to him via the special option in the dialogue wheel or if you have the noble knowledge inquisition perk.
Enchantress Ellandra can be recruited after completing the quest "My Lover's Phylactery". It may be possible to recruit her with the arcane knowledge inquisition perk but I do not know for sure. You can however recruit her by having Vivienne speak with her.
Smuggler Tanner can be recruited after you turn in the quest “Business Arrangements” by having Cassandra speak with them using the special dialogue option. Can also be done with Varric. Thanks Calex dEUS for this info.
Celmence is a tranquil that you can speak with during the quest "In Hushed Whispers " as far as I know there are no requirements to recruit him, simply talk to him.
Corporal Vale is located in the crossroads. After completing the Templar and Mage headquarters, the feed the refuges, clothe the refugees, and deal with the bandits side quests, as well as finding someone in Redcliff for him, you can go and speak with him and choose the recruit skilled refugees option to get an agent.
Siding with the Templars
If you chose to side with the Templars you can recruit Sgt. Barris. In order for him to survive through the mission you must not let the Hall defense gauge drop below 50% or he will die when the Templars go to destroy the barrier. Thank you Lliryn Dres for this information!
The Fallow Mire:
Sky watcher: is an Avaar that you can recruit after completing the quest "Lost Souls" Simply find him at the fort entrance and speak with him and he will join you.
Val Royeaux:
Belle is standing around in Val Royeaux and can be requited simply by speaking with her.
The Storm Coast:
The Blades of Hessarian Is kind of annoying to recruit. First you will have to make the Mercy's crest via one of the requisition quests given by the requisition officers at any of the camps in the zone. You will also need to pick up the quest "Cleaning House". Once you have both equip the Mercy's quest then head to the bandits camp. Once there you will be able to challenge the leader to a 1 on 1 duel. Upon defeating him simply speak another bandit and they will join you.
Crestwood:
Jana is located in Crestwood village in one of the houses. She wants to join the Grey Wardens. You can recruit her either by having Solas talk to her or if you have the Noble knowledge inquisition perk.
The Emerald Graves:
Fairbanks is recruited after completing the quest “Victims Of War” nothing else is required.
The Exalted Plains:
Loranil is available after completing the quest “By the Grace of the Dalish” In order to get the quest you need to do all of the Dalish quests in the area to earn their favor.
The Western Approach:
Frederick can be recruited after completing all of his quests including the slaying of a High Dragon. Once you have done that simply speak to him to recruit him.
Emrpise du Lion:
Michel de Chevin can be recruited after killing the demon Imshael and taking the keep that he is located in. Once you have done that simply speak to him and he will join you.
Skyhold:
Avaar Chief will come to skyhold and throw some animals against your castle walls in retaliation for killing his. He will be the first person that you sit and judge as the Inquisitor. If you choose to arm and exile him he will become an agent for you. If you choose any other option you will not be able to recruit him.
Servis can be recruited when you sit in judgement of him. You must speak to him in The Western Approach and bring him to Skyhold to do so. Simply order him to become a spy and he will become another of your agents.
Magister Alexius can be recruited during his judgement by ordering him to perform magical research for you. Any other option and you will not be able to recruit him. Requires the Arcane Knowledge Perk. Thanks Calex dEUS for this information.
Florianne After Killing Florainne in Halamshiral you will be asked to judge her corpse. If you choose to sentence her corpse to community service you will get an agent.
Lord Pel Harmond can be recruited by first speaking with Sera until she speaks about her concerns for the Verchiel region. This will open up the war table mission "A Battalion for Sera and Verchiel" after completing this mission at the war table go back and speak to Sera again. She will ask you to go with her to get paid. Agree to go and this will lead to a fight. After the fight is over Lord Pel Harmond will come out and you can speak with him. During the conversation you have the opportunity to recruit him or let Sera kill him. Be warned however that if you speak with him for to long Sera will automatically kill him. Thank you Soschiro for this information!
Jaws of Hakkon DLC:
Sigrid After completing the side quest "In Exile" you are given the option to either allow Sigrid to confess or to recruit her.
Storvacker First you need to complete the side quest "Storvacker Caged". Then after completing the main DLC mission you have the ability to judge Storvacker during which you are given the option to recruit him.
Thank you SilentlyForgot for this information about the DLC agents. Also thank you to punky0827 for confirming that the DLC agents do count towards the Persuasive achievement.
There may be more agents out there that I have not found yet, or failed to meet their recruitment requirements. I will update this as I find more agents that can be recruited. If there is anyone that you know of that I missed please leave a comment and I will update this list. Also if you know of other party members who can recruit a specific agent please let me know and I will add them also along with credit where it is due. I hope this helps those who need the 10 agents in a single playthrough.