Many popular website CMSs including WordPress, Joomla and Drupal were written in a programming language called PHP. PHP version 5 is about to reach end-of-life (EOL) and will stop receiving security updates in two months. Many WordPress and other PHP websites still remain on version 5.6 or older. Once support for PHP 5 ends in two months, such sites will become exploitable as new PHP 5 vulnerabilities emerge without security updates.
PHP 5 will be supported until 31 December 2018 and no longer be supported with security fixes, starting on 1 January 2019. That means that even if a vulnerability is discovered, it won’t be fixed by developers, leaving your website vulnerable.
PHP 7 has many improvements over PHP version 5. It's safe to upgrade to PHP 7.1 or 7.2. But since PHP 7.0 will also become end-of-life in one month, do not upgrade to PHP 7.0.
I mange several sites hosted by Bluehost and SiteGround. This site is hosted on Bluehost (VPS plan). Bluehost VPS supports up to PHP 7.2. But Bluehost's Shared Hosting plan supports up to PHP 7.0.
I contacted Bluehost's Support team to see when PHP 7.1 or PHP 7.2 will be available. According to Bluehost, they are working to update the PHP version on all BH accounts.
Currently we can increase the php version up to 7. We are aware that some of the customers wish to update the php version to 7.1 or 7.2, Our engineers are working to update the php version on all the bluehost account. We don't have an ETA yet, But soon it will be applied to all the Bluehost account.
Sooner or later, I think PHP version 7.1 or 7.2 will be available for all Bluehost accounts including Shared Hosting plan. (2020 Update: Now Bluehost also supports PHP 7.4.)
SiteGround supports up to PHP 7.3. And it's possible to apply different PHP versions on different (1st level) folders.
For your reference, if you sign up for a web hosting service by clicking the referral links on this post, I will get some commission. However, I do not recommend specific services just for commission. I've been using different webhosting services including SiteGround and Bluehost for years.
If you want a cost-effective webhosting service, you may consider Bluehost. If you sign up for Bluehost's Shared Hosting by clicking this link, you will be able to use Bluehost webhosting service with US$1/mo off.
※ Update in March 2019: Now Bluehost supports up to PHP 7.2 for all accounts (both with the new and the old cPanel interface.)
※ Update in August 2019: Bluehost added support for PHP 7.3.