Tiny core will need Firefox or some other browser installed. Puppy comes with SeaMonkey usually depending on what Puppy you downloaded (different breeds). Damn Small Linux comes with old Firefox. It is a dead distro though.
Not any new developments been going on with DSL.
I run AntiX and Puppy Linux (macpup,Bruno,NOP, Puppeee) depending on my gear and stuff. (pretty general, huh)
If you downloaded lupu Puppy 5.1. Your 128 mb of ram won't stand the strain. Tiny Core will boot though. Tiny core is Damn Small Linux revived by one of DSL developers. Read the Tiny Core Wiki and Forums on how to install Firefox, Desktops,use the package system it comes with etc.......
For puppy. You may need a smaller puppy download distro of 90 MB as Puppy runs in ram and will need it to finish booting.
Edit: forgot to mention. You will need to make a swap partition of lets say 512MB. It will improve booting and running a Linux distro.