I used this applet to figure out to some extent when the parameters are discrete. Here is what I found:

The group x/y/z is the (x,y,z) triangle group.
The numbers are listed so that x<=y<=z.
The element 1 is reflection in the side opposite to the angle labelled x.
The element 2 is reflection in the side opposite to the angle labelled y.
The element 3 is reflection in the side opposite to the angle labelled z

CONJECTURE 1: x/z/y is discrete and faithful iff neither of the two words 3231 or 123 is elliptic.

If you believe Conjecture 1 then the first word to go elliptic in the deformation of a triangle group is either 123 or 3231.

CONJECTURE 2: There is an infinite family of discrete but not faithful embeddings of x/y/z iff the element 3231 goes elliptic first.

*********************
Say that x/y/z is "good" if 3231 goes elliptic first. Otherwise call z/y/x "bad". Some general observations:
1. x/y/z good seems to imply x/y/(z+1) good
3. A neighborhood of infty/infty/infty is bad.

The list of good groups: (There might be some round-off error near the end of the 10's series.)

x/y/z x<10
10/y/z y<15
10/15/z z>15
10/16/z z>16
10/17/z z>18
10/18/z z>20
10/19/z z>23
10/20/z z>26
10/21/z z>30
10/22/z z>35
10/23/z z>43
10/24/z z>58
10/25/z z>110 (approximately)
11/11/z z>11
11/12/z z>13
11/13/z z>15
11/14/z z>18
11/15/z z>22
11/16/z z>20
11/17/z z>48
12/12/z z>16
12/13/z z>21
12/14/z z>32
13/13/z z>39