Front End Form with Dropdown in Matrix - Adding New
Your Code has two errors:
1.  was missing.
2. The way you were fetching options was wrong.

Here is the correct code:

&lt;input type="hidden" name="fields[emergencyContact]" value=""&gt;
&lt;input type="hidden" name="fields[emergencyContact][new1][type]" value="contact"&gt;
&lt;input type="hidden" name="fields[emergencyContact][new1][enabled]" value="1"&gt;
&lt;div class="grid-x grid-padding-x"&gt;
    &lt;fieldset class="cell small-3"&gt;
        &lt;label for="contactName"&gt;Name&lt;/label&gt;&lt;input type="text" name="fields[emergencyContact][new1][fields][contactName]"&gt;
    &lt;/fieldset&gt;
    &lt;fieldset class="cell small-3"&gt;
        &lt;label for="contactRelationship"&gt;Relationship&lt;/label&gt;
        {% set matrixField = craft.fields.getFieldByHandle('emergencyContact') %}
        {% set blockTypes = matrixField.getFieldType.getSettings.getBlockTypes %}
            {% for blockType in blockTypes %}
                {% if blockType.handle == "contact" %}
                     {% for field in blockType.getFields() %}
                       {% if field.handle == "contactRelationship" %}
                          &lt;select&gt;
                          {% for option in field.settings.options %}
                             &lt;option value="{{ option.value }}"&gt;{{ option.label }}&lt;/option&gt;
                           {% endfor %}
                           &lt;/select&gt;
                        {% endif %}
                      {% endfor %}
                 {% endif %}
              {% endfor %}
    &lt;/fieldset&gt;
    &lt;fieldset class="cell small-3"&gt;
        &lt;label for="contactPhone"&gt;Phone&lt;/label&gt;&lt;input type="text" name="fields[emergencyContact][new1][fields][contactPhone]"&gt;
    &lt;/fieldset&gt;
    &lt;fieldset class="cell small-3"&gt;
        &lt;label for="contactEmail"&gt;Email&lt;/label&gt;&lt;input type="email" name="fields[emergencyContact][new1][fields][contactEmail]"&gt;
    &lt;/fieldset&gt;


I hope it helps you.