At Shul I notice that some converts have names such as "Yoseph Ben Avraham V'Sarah" and some have just "Yoseph Ben Avraham". Are both correct? Is this something new (ish)?
Reform Judaism holds egalitarianism as a core value and calls everybody by both father's and mother's names (if both parents are Jewish). Converts are Ploni ben Avraham v'Sarah, but it doesn't stand out because everybody gets a matronymic. I've heard people talk about "Avraham Avinu v'Sarah Imeinu" but I have never seen somebody be called that.
Conservative congregations, in my experience, often use both names, but there are some that only use the father's name. (My Conservative weekday minyan recently merged into another, and in the process the default naming pattern changed.) Again, converts are no different from anybody else in this regard.
How long have they been doing it? I understand (anecdotally) that Reform has been using both parents' names since the 1960s or 1970s. I don't know about Conservative.
I don't have extensive Orthodox experience, but I don't think I've ever heard somebody called by both the father's and mother's name. This might happen in egalitarian or "partnership" minyanim; I've been to those a few times but can't recall what their naming practices were.
All of what I've said is for aliyot or other cases where you use a person's "usual" Hebrew name. A mi sheberach for healing uses the mother's name, universally in my experience. (I don't think I've heard both parents' names in this context in egalitarian congregations, even though you might expect that.)