Telecom operators worldwide are witnessing squeezed profit margins mainly due to hyper‐competition. Hence, new business models/strategies are needed to help operators reduce Operational and Capital Expenditures. In this context, the Network Function Virtualization (NFV) paradigm, which consists of running Virtual Instances of Network Functions (NFs) in Commercial‐Off‐The‐Shelf (COTS) hardware, represents a solid alternative. Virtual Network Functions (VNFs) are then concatenated together in a sequential order to form service chains (SCs) that provide specific Internet services. In this article, we study different approaches to provision SCs with resiliency against single‐link and single‐node failures. We propose three Integer Linear Programming (ILP) models to jointly solve the problem of VNF placement and traffic routing, while guaranteeing resiliency against single‐link and/or single‐node failures. Specifically, we focus on the trade‐off between the conflicting objectives of meeting SCs latency requirements and consolidating as many as possible VNFs in NFV‐capable nodes. We show that providing resiliency against both single‐link and single‐node failures comes at twice the amount of resources in terms of NFV‐capable nodes, and that for latency‐critical services providing resiliency against single‐node failures comes at the same cost with respect to resiliency against single‐link and single‐node failures. Finally, we discuss important insights about the deployment of bandwidth‐intensive SCs. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 70(4), 373–387 2017