Get a list of all the function signatures that have been compiled for a generic function

The MethodAnalysis package has tools for this and much more. For example:

julia> foo(x, y) = (x * y) ^ 2
foo (generic function with 1 method)

julia> foo(1,1); foo(1.0, 1);

julia> using MethodAnalysis

julia> methodinstances(foo)
2-element Vector{Core.MethodInstance}:
 MethodInstance for foo(::Int64, ::Int64)
 MethodInstance for foo(::Float64, ::Int64)

In some cases the specializations are not an svec, methodinstances makes it more straightforward to collect them in an easy-to-access format.

Perhaps there is a nicer way but this for sure works:

julia> foo(1,1);

julia> foo(1.0, 1);

julia> typeof(foo).name.mt.defs.func.specializations
svec(MethodInstance for foo(::Int64, ::Int64), MethodInstance for foo(::Float64, ::Int64), #undef, #undef, #undef, #undef, #undef, #undef)

Just to make sure it's mentioned here:

There is another package called LookingGlass.jl, which is quite similar to MethodAnalysis.jl, and provides a function for this too:

julia> foo(x, y) = (x * y) ^ 2
foo (generic function with 1 method)

julia> foo(1,1); foo(1.0, 1);

julia> using LookingGlass

julia> LookingGlass.func_specializations(foo)
Dict{Core.MethodInstance,Method} with 2 entries:
  MethodInstance for foo(::Float64, ::Int64) => foo(x, y) in Main at REPL[4]:1
  MethodInstance for foo(::Int64, ::Int64)   => foo(x, y) in Main at REPL[4]:1

I've wanted to merge LookingGlass.jl together with MethodAnalysis.jl for quite some time, but always too busy.

LookingGlass has other runtime reflection analysis tools, for modules, and functions, and methods, but I think the implementations in MethodAnalysis for this task are more robust, so @tholy's answer is probably best.