ADV Jirachi is most often seen as a Calm Mind user of some sort. The bulky Wishing variants are great Snorlax counters and initial switchins to Aerodactyl (resist to 3 of its 4 attacks, EQ isn't the end of the world). Psychic as strong, SDef-dropping STAB and then the last move can be anything; the three most common are Thunder, Reflect and Fire Punch. Other options include HP Grass, Ice Punch, Toxic and Protect; I posted about the benefits of the latter two
here.
Of course I can't not mention the more offensive variants, Superachi. Popularized by Jabba's team, it trades some of its bulk for speed and power while also picking up a third coverage move, making it much more dangerous right off the bat. While it's often used to lure a Dugtrio early for Porygon2 to revenge, it is at its most terrifying in the endgame against a scouted, weakened team.
It isn't always a CMer, though; Danilo went in-depth about his Wish + Protect utility set
here, I'm a big fan of it. Jira's also got some options with its signature move, Doom Desire; Triangles and I have had some fun results with a CB set.
I couldn't tell you a thing about DP Jirachi, but when I came around in Platinum I think it was most common as a Scarfer to abuse its sweet 60% Iron Head flinch rate and it's always remained a great mon, with its ability to revenge everything under the sun as well as Trick and U-turn yada yada. In the ScarfTar semi-stall era (where SubCM Jira was sometimes seen with lots of (Toxic) Spikes backing it up as the end-game sweeper, I forget whose team but the one with TrickScarf Starmie lead that was featured in the Smog is probably the best example), on teams like Megan Fox's and jumpluff/Earthworm's, a Bold Wishing Calm Mind set (like the one from ADV, yes) took everyone by storm, it was Latias' metagame and Jira was amazing against it. It always used Thunderbolt as the second move, but it alternated between the usual Psychic and the new Flash Cannon as STAB. While originally the set had its Defense nearly maxed out to do ridiculous shit like take 35% from Gliscor (very key in that meta) Earthquake, eventually it started getting faster; I remember Philip's Power Ranger's team had 308 Speed for Jolly Lucario. Neutral Salamence was all but gone from the meta by that point bar the rare bulky DDers but you wouldn't say no to beating them. Outrunning the surging-in-popularity offensive Suicune was also very key since if it's faster CM Jira is always a good check (until it Roars! but at this point it was all about the HP Electric since everyone was fascinated with this offensive Lucario check that also utterly destroyed Gyarados).
Another set that started gaining traction around this time was Substitute/Thunder Wave/Iron Head/Fire Punch. I remember matty's para team had it and iirc Pride/DJX09's suspect squad was packing it as well, it was the biggest nightmare on the planet. Kinda speaks for itself and it soloed many a team... it sorta died out once the dragons left but god what a terror it was/is. I don't think remember anyone running Body Slam but I'm sure someone was so annoyed by Gliscor that they made the change, although the tradeoff against Rotom kinda sucks.
Superachi started getting popular post-LatiMence ban, since Heatran was going from "very popular" to "the most popular" and HP Ground smoked it, ScarfTran was dying down at that time. No one expected its speed and power either, it was very good at cutting through the increasingly popular FWG cores (which lessened the need for Grass Knot since Swampert was almost nowhere to be seen and SDef Hippo wasn't seen too often yet) although it'd need to be wary around their best pal Scarf Flygon.
Lead Jirachi began to be a thing a little after that; I believe Earthworm used a 4 attack LO lead (who ran pretty much Superachi's set but with Iron Head > CM) to destroy Machamp while keeping the anti-Heatran thing going (there was a time when, like, Earth Power Shaymin wasn't too common and it also dunked on unsuspecting Tran). Pana used something similar, but he used Occa and SR as the 4th move (Machamp dying down or using Sash, ScarfTran resurging).
Pure utility sets, specially defensive with Wish/Protect/Body Slam/Iron Head, are not as common but they are sooooo good at shutting down offense, sorta for the same reason the Sub para set is. It's shut down by a lot of other defensive dudes but if those guys aren't around then everything is getting annoyed to no end. The bulk is immense and Protect makes it incredible at receiving Wishes, scouting choice dudes, and racking up every bit of Lefties recovery it can get.
In BW Jirachi started off the gen as a specially defensive bro to combat the terrifying Latios and LO Deoxys-S as well as the new Rankurusu (Reuniclus), it was also a great counter to the anti-rain champion Birijion (Virizion). It got ripped to shreds by newcomers Ferrothorn, Jellicent (~Nattorei Burungeru core never forget~), Poison Heal Gliscor, and old-timers Skarmory and bulky SD Scizor loved the free switch as well, but it was so good at its job that it remained a common set. Dissuading Garchomp, Randorosu/Landlos (Landorus) and Doryuuzu (Excadrill) switchins was so important that even rain teams used Body Slam on it (albeit sometimes with Thunder as well). undisputed created the Sub/CM Water Pulse set on his important rain stall, which turned the blobs + Ferro into setup fodder and trashed the Ground-types that normally screwed with Jira easily (Chomp eventually got banned).
After Exca was banned, Scarfers not named Politoed or Tyranitar started seeing the light of day again; forgive the bragging but Funkasaurus and I's
team was a big progenitor in showing the masses the usefulness of ScarfRachi. It was sooo good because it was such a reliable safety net against... almost everything, chipping all in sight with hazards + the momentum-keeping U-turn (forcing out Latios and retaining your advantage was just grogeous). It was an important member of many Hippo/Forre (semi-)stalls much like the aforementioned team. It remained a key figure all throughout the post-Genesect BW2 metagame (it was a SDef set a lot then and a while after because of the terrifying Tornadus-T). Despite other sets popping up - there was a period where Superachi was having some fun, and Shuca SR sets with vast coverage depending on its team needs (everything from Icy Wind to Energy Ball [not Grass Knot, Rotom-W eats that shit) to HP Fire to christ knows what else), Scarf has remained my favorite.