Steam login problems

It might help to reinstall your browser? I've never heard of such an issue, but it could simply be an problem with steam's account infrastructure. I've had the steam client itself show me as not logged in on the store before, I was then logged in after going to a different store page.

I'm actually getting the same problem... It doesn't open on either Chrome and Opera... IE seems to work fine though...

If you're having a problem trying to log in, try this steps:

Step 1: Close(or logout) your active Steam Client.
Step 2: Run Steam thru Big Picture Mode. If you can't do that, try dragging your Steam shortcut to your taskbar and right click Steam and select Big Picture.
Step 3: Login thru Big Picture Mode.
Step 4: Click the Power icon next to the Gear icon on your upper right of your screen and select Return to Desktop

I tend to have the problem that I'm logged in, but it says login for a while and then suddenly realizes I am already logged in...

There was a similar thread here, if you want to check if the OP found what the issue was.

I've been noticing weirdness with Steam too. Try going to the HTTPS community site and logging in there. I have the HTTPS-Everywhere add-on installed which loads the secure page by default, but I have to disable the add-on on some sites because it causes problems. I've noticed when I'm on non HTTPS steam sites that I'm not logged in, but when on HTTPS I am. From the non HTTPS site, it does exactly as you describe, it just re-directs me to the main store page.

Some other weird things I've noticed. If I have HTTPS-Everywhere disabled and go to the main store page and I have a notification (the green envelope at the top right), clicking on it gives me this error: Firefox has detected that the server is redirecting the request for this address in a way that will never complete. Enabling HTTPS-Everywhere fixes that problem, but it also causes some other minor issues on Steam.