sticky-search.user.js

// ==UserScript==
// @name         Stack Overflow Sticky Search Filter
// @namespace    http://tampermonkey.net/
// @version      0.0.0
// @description  Use localStorage to remember most recently used search filter
// @author       Patrick Roberts
// @match        https://*.stackoverflow.com/*
// @grant        none
// ==/UserScript==

(function() {
    'use strict';

    const tabsFilter = document.querySelector('.tabs-filter');

    if (tabsFilter !== null) {
        tabsFilter.addEventListener('click', ({ target }) => {
            if (target.matches('a[href]')) {
                localStorage.tabsFilter = new URLSearchParams(target.search).get('tab');
            }
        });
    }

    if ('tabsFilter' in localStorage) {
        for (const form of document.querySelectorAll('form[action="/search"]')) {
            const tab = document.createElement('input');

            tab.setAttribute('type', 'hidden');
            tab.setAttribute('name', 'tab');
            tab.setAttribute('value', localStorage.tabsFilter);

            form.prepend(tab);
        }
    }
})();