Unless a master of many styles monk, they don't interact at all; all other characters can only have one style active at a given time:
As a swift action, you can enter the stance employed by the fighting style a style feat embodies. Although you cannot use a style feat before combat begins, the style you are in persists until you spend a swift action to switch to a different combat style. You can use a feat that has a style feat as a prerequisite only while in the stance of the associated style. For example, if you have feats associated with Mantis Style and Tiger Style, you can use a swift action to adopt Tiger Style at the start of one turn, and then can use other feats that have Tiger Style as a prerequisite. By using another swift action at the start of your next turn, you could adopt Mantis Style and use other feats that have Mantis Style as a prerequisite.
Note that the wording in the Advanced Class Guide does not contradict this:
Style feats were introduced in Pathfinder Roleplaying Game: Ultimate Combat. The style feats presented in this chapter are based on fighting styles more appropriate to the brawler's rough-and-tumble combat disciplines.
As a swift action, you can enter the stance employed by the fighting style that a style feat embodies. Although you cannot use a style feat before combat begins, the style you are in persists until either you use a swift action to switch styles or the end of the combat encounter, whichever occurs first. You can use a feat that has a style feat as a prerequisite only while you are in the stance of the associated style.
Weapon Style Mastery feat allows one weapon style (a style feat that lists Weapon Focus as a prerequisite) and a second style to be active at once. That means that you don't need master of many styles monk to combine them, though they'll need the Martial Focus feat if they don't get weapon training class feature.
Yeah, but assuming you've picked an option that lets you stack them (Master of Many Styles Monk, Freestyle Fighter, Weapon Style Mastery), how do they interact? Do you get increased dice on the first hit, or no? Do you get temple sword Pummeling Style attacks?
Pummeling style is out do to their adding the line "This ability works only with unarmed strikes, no matter what other abilities you might possess" to the feat.
Someone said that since it says "up to 1.5 str" that it doesn't do anything if you're two-handing the weapon, but if you're one handing then it works like normal.