Waterproof Fountain Pen Inks
Most fountain pen inks are not super water-proof. There are some that are more waterproof than others, but most have some limited resistance. That is, if you spill coffee on your next brilliant idea and wipe it up, it should still be legible. If you are caught in the rain, it won't disappear completely. Some inks like Noodler's Bulletproof inks even protect against bleach and other craziness, but for day to day living and writing, most inks should be fine, especially if you are writing in a notebook or journal, or on papers that most likely will not be taken to the bathroom or outside in a storm. If you leave your paper under a running tap for several minutes, or even soaked in water for a long time, you may begin to find your ink disappearing beyond legibility.
The sample above looks a little more dramatic because I used a flex nib that left a lot of ink on the page and I was too impatient to wait more than a few seconds for it to "dry." However, if your writing is dry and there's not too much ink, your writing should generally fare a little better. Often the top layer of ink that hasn't bonded to the paper will be what washes away, and there will still be writing left. There are two other major factors in water-resistance-ness. One is how long you've left it to dry - the longer, the better. The inks below in the sample were left to try for about 30 seconds. If they were left for a little longer, or even overnight, or best still for a week, some of them may have fared a little better. The type of paper you use will also be important. If your paper has some coating on it (think greeting cards or postcards), the ink will have a much harder time bonding with the paper fibres below. However, if you are in the market for a particularly water-resistant or waterproof ink, here are a few to consider. There are iron-gall inks like Rohrer & Klingner's Salix or Scabiosa, Platinum Blue-Black and Diamine Registrar's Ink. Really old school iron gall inks can be corrosive in your pens, but most modern day iron galls are fine with a little bit of care (not leaving the inks to sit in your pen for too long, as with all other inks).
There are also Noodler's "Bulletproof" Inks, including their standard Black and 54th Massachusetts. Some of Noodler's other inks will be partially waterproof - especially the ones that have a Bulletproof component - for example their Blue-Black will have some of the blue wash away, but the black will remain. Other notable ones include the Bad Blue Heron, Bad Black Moccasin, X-Feather, Lexington Grey and #41 Brown. Last are the carbon inks - made by Sailor in Kiwa-Guro black and Sei-Boku blue-black or Platinum's Carbon Black. These inks are made from micro fine carbon pigments - fine enough to make it through your feed without clogging, but don't leave it in there too long! Many, many other inks show good water-resistance, including J. Herbin Perle Noire, Sailor Jentle Black, Sailor Jentle Blue...the list goes on. If you are interested in particular ink you can always Google to see if it will be water proof, but remember that most inks will be alright with a few drops anyways.