When deciding if a blog for T.LY link shortener, should be on a separate domain, I reached out to IndieHackers to get their opinion. I originally was going to use my tdotly.com domain for the blog and marketing site but decided to just use blog.t.ly and /blog redirects on T.LY from all the feedback. This is primarily for SEO reasons to consolidate authority to a single domain. Google claims that sub-domains are not worst/better compared to a subfolder if this is easier for your server setup. Just make sure you add both sites to Google Search Console.
It was pretty clear that /blog on your primary domain is the best option but this is also the most complicated due to routing if your blog is hosted on a CMS like WordPress. One solution is to use Cloudflare as a reverse proxy using their cloud workers.
What are your thoughts on the best URL for a blog?