This question coming up a million times bugged the hell out of me when I was waiting on my account, so I'll give a full (as far as I'm capable) answer.
Spinoffs aside, there's three main streams of MH games. Each has a pattern. I think they're made by different teams, but don't quote me on that.
Mainline series
1, G, 2, 3, and soon 4.
All have online multiplayer. All on console until 4, which made a generation transition onto 3DS.
Until 4, they were all on the biggest console of the generation (PS2, Wii). 4 is on the biggest handheld instead (3DS).
No concurrent multiplatform releases.
Only G has been ported. The first time was the start of the Portable series, the second time it was a 4 year late port from one gen to another (PS2 -> Wii).
Portable series
P, P2, P2G, P3, 3G.
All have local multiplayer. All on handheld.
All on PSP until 3G, which made a generation transition onto 3DS.
No concurrent mutliplatform releases.
P3 and 3G were ported only to the console that could link with them and provide online play (PSP->PS3, 3DS->WIU).
Frontier
S1 through S10, F1 through F5.
Online multiplayer. Free to play or pay to play depending on region.
Originally PC exclusive. 360 versions were announced 3 years in.