Usually when I design pages I design for about 6 different resolutions and some smaller phone screens slip through the cracks but it's generally small screens x2, bigger phones, tablets, laptops and then everything bigger. On my laptop I can click in the search and get the advanced search options (didn't know that until today so it shows how often I search), but not sure if your screens are higher res than mine therefore giving different results. I could look at the code but honestly I can't be fucked.
I see that in Firefox which I usually use, but if I switch to Google Chrome then I just see recent searches as @Einherjar86 described. Maybe a recent update to Chrome has a disagreement with how the advanced search popup works.
