I see that you can save searches on android but can't save them on the web app. Am I missing something?
Unfortunately, this feature is not available on the web-app right now, but you could save one specific search which would become your default start page on he web-app. To do so, go to the Settings (gear icon in top-right corner) -> General Settings and choose "Custom query" from the drop-down menu in the "Start page" section.
In a separate thread, where I commented that saved searches were not available in Todoist, this is where I got my information. I went to Support, searched for Saved Searches and this is the page that came up. It would be good to put the correct and up-to-date information in this thread!
Thank you for clarifying that. Basically, the status hasn't changed since my last reply, though. Saved searches are still not available on the web, only on mobile, but we're working on this right now.