Snub AntiPrisms
This is an infinite family of 2uniform polyhedra where a base polygon (an ngon) is surrounded by vertices of the form n.3.3.3.3, the snub vertices are then of the form 3.3.3.3.3 Substituting 2 for n gives the Johnson Solid snub disphenoid, {3} is a regular icosahedron, {4} is the Johnson Solid snub square antiprism and {5} is a snub pentagonal antiprism, however this last example is noticeably not convex. Note: The ngonal members of this family are briefly discussed in Professor Bonnie Stewart's "Adventures Among the Toroids" 2^{nd} Ed. Ex 115 on page 160.
Substituting an n/dgon for the ngon in the above extends the family to allow an infinite number of locally convex polyhedra (in the sense that no dihedral angle is > pi). If this is not obvious from the images above take a look at this cutaway model of a 7/3 snub antiprism where only one rotation of the triangular faces around the prismatic axis of symmetry is shown.
These polyhedra are generated from the antiprisms by dividing the antiprism into two portions, each of which consists of the base n/dgon and it's edge connected triangular faces. A ring of snub triangles is then inserted between these two portions. This is evident in this example of a highlighted 5/2 snub antiprism with the snub triangles in yellow.
The upper bound for local convexity is at n/d = 4.7445... Click here for proof. Cases generated around this limit include 4.5 (see 9/2), 4.666 (see 14/3) and 4.75 (see 19/4). Only the first two are convex.
A full table of locally convex snub
antiprisms with d>1 and n<=12 is below.
Snub Antiprisms:  5/2

7/2

7/3

8/3

9/2

9/4

10/3

11/3

11/4

11/5

12/5

Snub antiprisms also exist for n/d<2. The lower limit is undetermined. In these cases the base polygon is retrograde. In general the retrograde n/(nd) snub antiprism is isomorphous to the n/d snub antiprism. Examples of retrograde snub antiprisms are given here: 7/4, 5/3, 3/2, 10/7, 4/3, 5/4 ^{[1] }
As the icosahedron is a member of this family, the various isomorphs to the triangular snub antiprism also occur in this list of isomorphs to the icosahedron.
The snub antiprisms can be gyroelongated by the insertion of a band of triangles between the two halves of the snub antiprism. This generates a family I term the pentakis pentaprisms. They can also be deformed into antiprisms and gyrobicupolas  see 'Twisters'.
A family also exists of "great" snub antiprisms, an example of the 7/3 great snub antiprism is shown above. These polyhedra are isomorphous to the snub antiprisms but have starred vertices. They have the same relationship to the snub antiprisms as the great icosahedron has to the icosahedron. Indeed the great icosahedron is a member of this family and can be regarded as a "triangular great snub antiprism".
A
table of those generated
to date with n/d>2 is as follows:
Great Snub AntiPrisms:  4

5 ^{[1]}  5/2

7/2

7/3

8/3

9/2

9/4

10/3

11/3

11/4

11/5

12/5

Great snub antiprisms exist for n/d<2, the retrograde base polygons then having the effect of partially uncrossing the starred vertices. Examples are the 3/2, 4/3, 5/3, 7/4, 8/5 and 5/4 ^{[1]. }Note the triangular edges on the 4/3 example are coplanar with the square faces.
Exotic Snub AntiPrism Isomorphs
Two additional isomorphs exist for certain ranges of n/d. These are (a) the hybrid snub antiprisms and (b) the inverted snub antiprisms. Both families have pyramidical rather than prismatic symmetry and are 4regular polyhedra.
The hybrid snub antiprisms are so named as they have the appearance that one half of a prograde n/d snub antiprism is joined to one half of its retrograde n/d* twin. For example the 5/25/3 hybrid (above) the 7/37/4 hybrid or the 8/38/5 hybrid. The limits of this 'hybrid' family are undetermined. The hybrid 33/2 snub antiprism also exists and is synonymous to a triinverted icosahedron. (see isomorphs to the icosahedron). This is apparent by examination of this 44/3 hybrid.
The inverted snub antiprisms are so named as one of the half snub antiprisms is inverted back into the centre of the figure, meaning that the {n/d}gonal cap is predominantly hidden. For example this 4, 5/2 or 7/3 example (above). The limits of this 'inverted' family are undetermined. The inverted triangular snub antiprism also exists and is synonymous to a trieverted great icosahedron. (see isomorphs to the icosahedron)
An unexpected coplanar case ^{[2]}
With the size of the base polygon, the two bases move closer together, at n=9 they have reached a point where the edge connected triangles are coplanar with the base, the enneagonal snub antiprism is shown on the left above (VRML, OFF). Coplanarity is also evident in the 9/7 snub antiprism (above right) (VRML, OFF) where the edge connected triangular faces are coplanar with the 9/2gonal caps but are now pointing inwards. (Note: the 9/2gons in this model have been coloured blue to avoid migraines when viewing the VRML file).
Notes and acknowledgements
[1] I am indebted to Adrian Rossiter, developer of the Antiprism software for the generation of the great 5, the 5/4 isomorphs, and the generation of the coplanar 9 and 9/7. Adrian has developed software to generate an arbitrary isomorph of any n/d snub antiprism and has also generated two fascinating animations of the {101/d} antiprism for varying d. The files are around 6MB each, see www.antiprism.com/misc/snu101_s0.gif (snub antiprisms) and www.antiprism.com/misc/snu101_s1.gif (great snub antiprisms).
[2] The coplanarity of the {9/7} snub antiprism was discovered by Roger Kaufman, that of the {9}snub antiprism by Adrian Rossiter.