There are not so many open source survey software. Many have tried to emerge, but only LimeSurvey and Formbricks succeed. They are the most powerful open source survey software available today. If you look at more than a full dedicated survey software (ie. CMS or Community Management software), look at Drupal or WordPress. They have some nice webforms & survey abilities.

LEADING OPEN SOURCE SURVEY SOFTWARE

LimeSurvey basically contains everything you need for doing nearly every survey with grace. LimeSurvey (former PHPSurveyor) has been used for years by thousands of web sites, and is the leading open source survey software.
 
Formbricks is an open-source (AGPLv3) survey platform built to collect feedbacks. With Formbricks, you can replace many existing survey tools: 
  • Standalone surveys (share via link): Replace Google Forms, Typeform or any other link survey tool with Formbricks Form Builder. Use lots of question types and comprehensive customizations,
  • Scalable website surveys: Formbricks lets you run well-timed and anonymously targeted surveys on any public website. 
  • Highly targeted app surveys, enriched with profiles with attributes and specific actions. You can build cohorts for highly targeted in-app surveys. 
Formbricks survey platform is mostly free, even for commercial use. Over time, all survey features will stay part of the free Community Edition.

OTHER OPEN SOURCE SURVEY SOFTWARE

 
This module adds a webform nodetype to your Drupal site. Typical uses for webform are to create questionnaires, contact or request/register forms, surveys, polls or a front end to issues tracking systems.
 
Contact Form 7 is the historical WordPress plugin designed for surveys. It employs a sophisticatedly modularized architecture and its original Schema-Woven Validation technology.
 
WPForms is a drag & drop WordPress form builder with a very neat UX. One can create contact forms, feedback forms, subscription forms, payment forms (including Stripe, Square & PayPal), and other types of forms very quickly. With more than 6 millions downloads, WPForm is one of the leading solution in the WordPress space.
 
A quite new plugin, proposing a very intuitive drag-and-drop interface enabling you to design online forms like surveys, polls, quizzes, registration, payment, lead, email forms, and calculator forms.