With around 1.2 million apps currently available in the App Store, actually finding what you’re looking for is a lot harder than it should be.
Sure, it’s pretty easy to find those brand name apps, like Twitter or Instagram - but unless the one you want has made it to the top of the category, been featured or you know the exact name of what you’re looking for, things can get tricky.
At the moment there’s doesn’t seem to be one ‘best’ way to find apps, so I’d like to outline a few different ways that have worked for me.
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1. Use Google

It seems a little ironic, using Google to search Apple’s database, right? But a few handy Google operators might remind you why this is still the world’s favourite search engine…
Here’s an example search for social media apps
1. Google: site:itunes.apple.com/gb/app “social media”
2. Scroll through the results and click on a link to be taken to the iTunes link for the app
3. (optional, but sometimes useful) you can use the ‘Search tools’ feature under the browser bar to order your results by date, or to search for apps that came out within the past year, or month.

How it works

The ‘site:’ operator ensures that Google searches only site:itunes.apple.com/gb/app– which is the app section of the UK iTunes
Whereas “social media” (parentheses included) will give you pages that match the exact term ‘social media’)
So, for example if you wanted to search for apps that mention the phrase 'photo editing' you'd write...
site:itunes.apple.com/gb/app “photo editing”
You can also remove the ‘gb’ to make a search across the entire app store – so that search would now be site:itunes.apple.com/app “photo editing”

2. Ask in Apple Discussions

Apple discussions is a great place to find a lot of often helpful, often vocal Apple users.  And, in terms of finding what you want, sometimes it's best to save yourself the time searching and ask for a personal recommendation.
So - for example, if you wanted an advice on apps that are best for managing your social media you could ask on the forums.
There's two ways to do this, you can choose the section you want to post in, or pose your question, Google style, in the search bar.  The second option will pull up similar threads, if they exist, and/or suggest a good section for you to post your search in

3. Use Vionza

Vionza provides a somewhat less overwhelming way to carry out a search of the App Store. It's also got some filtering around factors like price and rating, along with the ability to arrange results.

Here’s how to use it:

2 Change the flag at the top right hand corner to 'UK Store’ if it's not already (this will ensure that it searches the UK App Store)
3 Enter your keywords and choose the category you want to search in, then click search
4 If necessary, arrange your results using price and rating filters on the left side of the page.

4. Use Fnd

Like Vionza, Fnd is another website that searches the app store for you. Fnd has a slicker interface, but seems to lack the filtering that Vionza offers - so I like to think of it as a second choice, if I can't find what I'm looking for in Vionza.
1 Navigate to https://fnd.io/
2 Change the flag at the top left corner to British if it’s not already (this will ensure that it searches the UK App Store)
3 Enter the keywords you wish to search with
4 Choose the ‘apps’ tab from your search results
5 Scroll until you find something that interests you
6 Click ‘view in app store’ to go the app's page in the app store

5. Be Persistent!

If you’re searching within the App Store itself, often it pays to be persistent and try a variety of methods…

Try a variety of search terms

For example - if you were looking for 'social media' related apps,  you might want to try related keywords such "social networking" "social media management" or even something that seems redundant like "social media app" . There's often very little logic to it, but when it comes to searching, persistence pays.

Try the developer’s name

If you know the developer creates good apps - and that they create apps in the category you're looking for, try searching by. Often, the developer’s is specific enough to come up as an exact result.

Use ‘App Discovery’ sites / apps

There's a number of third party websites and apps that are geared around covering the immense app store marketplace. They're more oriented towards discovering new and on-sale apps, and not so good at direct search.
This is usually a last option for me as it takes longest and feels like I'm meandering at this point.
However, if you have the time/inclination to browse through a bunch of curated categories - try a few of the sites/apps on this Alternativeto list.

App Store changes in iOS 8

We know that iOS is going to add additional features, which we're hoping will make things easier. Here's a few things to expect in iOS 8's App Store.
Explore Tab - The Explore Tab bar sits at the bottom of the App Store app and provides a more streamlined means of browsing the App Store's various categories and subcategories - at current these can only be accessed via the Categories option at the top of the Featured tab.
Search Tab - Changes to search will show you, Twitter style, what all the current trending searches are - showing you what's popular and what apps people are talking about.
Related Search Terms - App Store will now show you search terms related to what you just searched for. This should hopefully save you some time coming up with alternative search terms if you can't find what you want, making your search easier.
Scrolling Search Results - iOS8 will return a continually scrolling list of results, including app information and screen shots for each one. This is opposed to you having to swipe across each one, like in iOS7 - which should save you a lot of swiping!
Editor's Choice Icon - Any app that has ever made 'Editors Choice' will be identifiable with an Editor's Choice icon - this way you know that at one point this app was good enough to be picked out by Apple, even long after it's been highlighted.