As requested, here a full list of all the project badges possible, shown as bronze.
ESP LLR (Extended Sierpinski Problem)
PPS LLR (Includes both PPS and PPSE -- Proth Prime Search [Extended])
PSP LLR (Prime Sierpinski Problem)
SoB LLR (Seventeen or Bust, aka The Sierpinski Problem)
SR5 LLR (Sierpinski/Riesel Base 5)
SGS LLR (Sophie Germain Search)
TPS LLR (Twin Prime Search) (retired)
TRP LLR (The Riesel Problem)
321 Sieve (suspended)
Generalized Cullen Woodall Sieve (suspended)
ESP/PSP/SoB Sieve (Previously called the PSP Sieve) (suspended)
AP27 (Arithmetic Progression of primes - 27)
Genefer, aka GFN -- Generalized Fermat Number prime search (including N=15 through N=22)
PSA (Project Staging Area): PRPNet and Manual Sieving
This list does not include the "one-off" badges that are awarded to the winners of the annual "Tour de Primes" competition.