The Filter Menu, overall, does not have a huge impact on your search engine rankings, compared to something like external link building. Don't expect to install it, and for you to be ranking number 1 overnight, or likewise to suddenly drop down in the rankings. 

The reason for this is that the Filter Menu uses the existing URLs and pages that are already present in your store and theme, or if it adds new pages, these are not likely to have many inbound links.

That said there are some interesting cases to cover, and ideas for optimizing how search engines view your site.

As always this is general advice, and you should always speak to your SEO professional before making any changes.
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Often the Filter Menu will create new collections to enable its advanced filtering. These pages typically won't have many incoming links, so search engines won't typically be very interested in these new collections. However, they can be used to your advantage for long-tail searches. 

Long-tail search is the sum of the infrequent search terms. Any single search term or resulting page is rarely searched for or seen, however, if you get a lot of them summed together, then this can make up a substantial amount of search traffic. 

This means that having more collections is not a bad thing, and in fact can contribute to significant long-tail search traffic.
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When you start to apply filters, a new page is dynamically created with those specific filters applied. Depending on how your theme works, these can have the same title and header as the base collection, or a new title is generated that is, perhaps, not very customer and search engine friendly.

For example, if you go to a page with say Shirts, then apply a filter for the color Red, most themes with say one of two things in the page header:

Ideally, we want it to say Red Shirts. Not only is this good for your customers, but this is also good for search engine phrases. 

When you install the Filter Menu, a helper is also added to your theme code, this helper generates nicer titles, you can use this helper in your H1  and title  tags by using the following liquid code:

{% include 'pt-collection-title' %} 

instead of collection.title or other title code. 

Search engines will see these as unique pages if you update the H1  and title tags, so that they provide some value as a search result listing.
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Canonical tags can be used the other way so that instead of creating pages with value when the filters are applied, we assign all the value to the base collection page. Any pages with filters applied are marked as being the exact same page as the page without any filters applied, as far as the search engine is concerned. 

We have a helper for applying a canonical tag to your theme, you can run it from the theme installer page in the Power Tools app, you can find this in the app sidebar under Theme Installer -> Filter Menu -> Additional Helpers -> Canonical tag shows 1 filter, or here is a direct link.

Should you use a canonical tag or not? That is the tough question, and the answer is that it depends, and you should speak to your SEO professional.

Our typical recommendation is to not use canonical tags and use better titles that are unique. However, this may not work in all situations.

We do not recommend using noindex on the Filter Menu links, as a canonical tag works better for optimizing your internal graph. 

Shopify always comes with a robots.txt file that cannot be altered. It is optimized for search engine crawling. The Shopify provided robots.txt file will block any URLs with a + symbol in them, so any page with more than 2 filters will be automatically blocked from being indexed by search engines.