There are a few websites with lists and/or databases of graphs, maps and polytopes.

- [House of graphs][1] has a searchable database of interesting graphs and aims to serve as a repository for lists of graphs and graph generators,
- [Encyclopedia of graphs][2] is an online encyclopedia of graph collections with some data about different families of graphs,
- [Database of graphs in combinatorica format][3],
- [Encyclopedia of Finite Graphs][8] is a "set of tools and data to compute all known invariants for simple connected graphs".

There are several censuses of symmetric objects (for example [Marston Conder][4], [Dimitri Leemans][5], [Primož Potočnik][6]).

I would also like to know about other websites of similar nature out there, since I'm working on one myself. It's still an early stage [project][7], but has some use already. It aims to one day be able to provide a platform for computational attempts for all sorts of discrete objects (you can see some features for the exploratory aspect on the website).

  [1]: https://hog.grinvin.org
  [2]: http://atlas.gregas.eu
  [3]: http://www3.cs.stonybrook.edu/~skiena/combinatorica/graphs/
  [4]: https://www.math.auckland.ac.nz/~conder/
  [5]: https://www.math.auckland.ac.nz/~dleemans/resources.html
  [6]: http://www.fmf.uni-lj.si/~potocnik/work.htm
  [7]: http://www.discretezoo.xyz
  [8]: https://github.com/thoppe/Encyclopedia-of-Finite-Graphs